Statistical blending of weather data sets

ABSTRACT

In an approach, a method for fusing point data with areal averages is performed by a computing system. The fusion procedure is coherent, in the sense that the computing system takes into account what the areal averages represent with respect to the point data. The overarching goal is to fit a model that takes into account the information derived from both data sets. The areal averages provide an estimate for what the integral of a model representing the behavior of the environmental variable should be over a particular district and the point values indicate the estimated value at particular locations. Thus, the integral of the fitted model over a district of the grid should approximate the value provided by the areal averages while also approximating the value provided by the point data for locations which are provided by the point data.

COPYRIGHT NOTICE

A portion of the disclosure of this patent document contains materialwhich is subject to copyright protection. The copyright owner has noobjection to the facsimile reproduction by anyone of the patent documentor the patent disclosure, as it appears in the Patent and TrademarkOffice patent file or records, but otherwise reserves all copyright orrights whatsoever. © 2016 The Climate Corporation.

FIELD OF THE DISCLOSURE

The present disclosure relates to computer-based systems that areprogrammed for blending together multiple different weather data sets ina statistically sound manner. More specifically, the present disclosurerelates to using computer programs to blend together weather data setsthat have different support, such as blending together point data withareal averages.

BACKGROUND

The approaches described in this section are approaches that could bepursued, but not necessarily approaches that have been previouslyconceived or pursued. Therefore, unless otherwise indicated, it shouldnot be assumed that any of the approaches described in this sectionqualify as prior art merely by virtue of their inclusion in thissection.

Weather forecasting is the application of science, technology, andstatistics to predict the state of the atmosphere for a given locationat some future point in time. The endeavor to fully understand Earth'sclimate system and to predict the weather has been a goal of humanityfor millennia. Weather forecasts are typically made by collectingquantitative data about the current state of the atmosphere at a givenplace and using the data to drive a simulation or physical model of theatmosphere to predict how the atmosphere will change over a given periodof time. For example, identifying changes in environmental variablessuch as temperature, air currents, barometric pressure, moisture, and soforth.

The collection of the quantitative data is performed by using varioustools, such as satellite image data, weather stations, temperaturereadings, humidity detectors, and so forth. Physical models used forweather forecasting decompose the earth (or other geographical region)into a uniform grid where the various environmental variables are givena particular value at each location within the grid. The physical modelthen runs through the grid simulating the physical processes that causechanges in weather over time to reach a future predicted state. However,the base data collected from the various weather stations do not fullycover the grid, nor are the readings taken at uniform times or withtools that have identical measurement errors. In fact, in most casesthere are far more points on the grid where the environmental variablesare unknown than known. As a result, to convert the base observationsinto values for each point on the grid, a process known as dataassimilation is performed which uses a combination of information tofill in the points where observations are not explicitly available. Theresult is a value for each of the environmental variables for each pointin the grid, which is collectively referred to as an “analysis”. Theanalysis is then used to set the initial state of the physical modelsimulation which is stepped forward in time to predict the weather atsome future time. In some cases, a forecast model is also used to fillin the informational gaps, which is referred to as an analysis/forecastcycle. In essence, an initial condition is set by an analysis, aforecast is run from the analysis, and the forecast is then used to fillin or smooth out the gaps in the next analysis in a repeating cycle.

Since the initial condition of the atmosphere generated by the analysisis uncertain due to the observational data being incomplete, climatescientists will often run forecasts using a set of different initialstates based on the known or estimated error of an analysis. Theresulting forecasts, each representing the future state of theatmosphere assuming that the values of the environmental variables inthe grid were in a slightly different initial state is referred to as aforecast ensemble. The overall behavior of the ensemble, rather thansimply one forecast, is then used to better capture the uncertainty inthe forecast.

In most cases, the analyses are performed by government agencies (and insome cases private agencies) and made available via various databases,for example the U.S. Climate Forecast System (CFS), the European Centrefor Medium-Range Weather Forecasts (ECMWF), and so forth, which provideanalyses that can be accessed and used for research by weatherscientists. These organizations often provide analyses at differentgranularities of time, such as six hours, daily, weekly, and so forth,as well as at different geographical granularities (for exampledifferent grid sizes).

Data assimilation techniques and forecasting models constantly evolveover time as atmospheric scientists develop a better understanding ofthe physical processes governing atmospheric evolution. As a result, ifone were to view the analyses taken by various public and privateorganizations over an extended period of time (for example the lastthirty-forty years), the changes in the data assimilation technique orforecasting model used can have a drastic impact on the analysis and theresulting forecast. To combat the non-uniformity of the techniques usedto create the original analysis, climate monitoring organizations willoften go back to the original observation data collected over a pastperiod of time and apply a consistent data assimilation technique(usually one that is more up-to-date than the original technique) fromthat past period of time to the present. As a result, theinconsistencies are removed and the skill of forecasting models can bemore easily evaluated. An analysis that is produced in this manner isreferred to as a “reanalysis” since the data is being reanalyzed using aconsistent technique.

Evaluating the skill of a forecast model requires a significant amountof forecasted predictions and corresponding observations with which tocompare those predictions. However, when testing a new model, it isimpractical to train on historical observation data and then evaluate atsome point in the future based upon the analysis generated at that time.Especially for longer range forecasts, it might take over a month beforea given forecast can be evaluated, and years or decades before enoughdata can be collected to tell whether the forecast model is actuallyskillful. As a result, climate scientists often perform “reforecasts”,which is a forecast based on past analyses (or more preferablyreanalyses). For example, if a reanalysis covers the past 30 years, aforecast model can be initialized from those conditions and used toproduce simulated forecasts across the 30 year period. Thus, areforecast provides evidence of what a forecast model would predict ifit had been used to forecast environmental conditions at some previouspoint in time. As a result, the predictive skill of the forecast modelcan be evaluated at a variety of leads by comparing the predictions tothe corresponding observed environmental conditions at that time.

SUMMARY OF THE DISCLOSURE

The appended claims may serve as a summary of the disclosure.

BRIEF DESCRIPTION OF THE DRAWINGS

In the drawings:

FIG. 1 illustrates an example computer system that is configured toperform the functions described herein, shown in a field environmentwith other apparatus with which the system may interoperate.

FIG. 2 illustrates two views of an example logical organization of setsof instructions in main memory when an example mobile application isloaded for execution.

FIG. 3 illustrates a programmed process by which the agriculturalintelligence computer system generates one or more preconfiguredagronomic models using agronomic data provided by one or more externaldata sources.

FIG. 4 is a block diagram that illustrates a computer system upon whichan embodiment of the invention may be implemented.

FIG. 5 illustrates a functional overview of a data blending subsystemaccording to an embodiment.

FIG. 6 illustrates an example process flow for blending point data andareal averages in block diagram according to an embodiment.

FIG. 7 illustrates how to use a fitted model to generate a prediction ofan environmental variable at a given time and location in block diagramform according to an embodiment.

FIG. 8 depicts an example embodiment of a timeline view for data entry.

FIG. 9 depicts an example embodiment of a spreadsheet view for dataentry.

DETAILED DESCRIPTION

In the following description, for the purposes of explanation, numerousspecific details are set forth in order to provide a thoroughunderstanding of the present disclosure. It will be apparent, however,that embodiments may be practiced without these specific details. Inother instances, well-known structures and devices are shown in blockdiagram form in order to avoid unnecessarily obscuring the presentdisclosure. The description is provided according to the followingoutline:

-   -   1.0 General Overview    -   2.0 Example Agricultural Intelligence Computer System        -   2.1 Structural Overview        -   2.2 Application Program Overview        -   2.3 Data Ingest to the Computer System        -   2.4 Process Overview—Agronomic Model Training        -   2.5 Data Blending Subsystem        -   2.5.1 Data Blending Subsystem Functional Overview        -   2.6 Implementation Example—Hardware Overview    -   3.0 Example System Inputs    -   4.0 Gaussian Model        -   4.1 Notation        -   4.2 Temporal Variability        -   4.3 Spatial Variability        -   4.4 Multivariate Model        -   4.5 Point Measurements v. Areal Averages        -   4.6 Model Summary        -   4.7 Covariance        -   4.8 Initial and Posterior Mean States        -   4.9 Log-Likelihood Function        -   4.10 Missing Observations        -   4.11 Hindcasts and Reforecasts        -   4.12 Backward Smoothing        -   4.13 Backward Sampling    -   5.0 Data Blending Process Flow    -   6.0 Prediction Process Flow    -   7.0 Use Cases    -   8.0 Extensions and Alternatives    -   9.0 Additional Disclosure

1.0 General Overview

The problems of uncertainty quantification and data fusion with changeof support (for instance, blending point data with areal averages) areubiquitous in weather and agronomic research. For example, manyapplications of weather analysis, such as predictive modeling, requireprobabilistic estimates of historical and forecast weather blended froma multitude of sources. One major challenge is to make the most of themany data sets available for each weather variable. First of all, eachvariable has its own spatial and temporal extent, support (for examplepoint data vs. areal data), and resolution. For example, reanalyses andsatellite data are typically gridded areal averages, whereas weatherstation data represent point data that is irregularly distributed inspace. Moreover, each data set has its own level of accuracy, which mayor may not be available to researches. However, by comparing multipledata sets, the biases inherent to those data sets can be quantified, theinstrument precision can be estimated, and the data set(s) mostappropriate for a specific purpose can be assessed.

In an embodiment, a scalable Bayesian approach is applied that isagnostic with respect to the type of data being modeled, as well as thespatial and temporal resolution. The approach relies on discrete processconvolutions to produce a smooth spatial interpolation over the domainof interest, which can be approximated by local second-degreepolynomials to estimate areal averages. To capture temporal variability,a first order Markov process is specified for gridded baselines andseasonal components. The result is a state-space model with basicforecasting ability. The approach described herein does not requirecostly manipulations, such as inversions or factorizations, of largematrices and can account for location-dependent and time-dependentinstrument precision, as well as missing values.

In an embodiment, a method for fusing point data (such as weatherstation measurements) with areal data (such as reanalyses and/or remotesensing products) is performed by a computing system. The fusionprocedure is coherent, in the sense that the computing system takes intoaccount what the areal data represents with respect to the point data.For example, areal data represents the average value of an environmentalvariable across a district of a spatial grid. The grid may represent anyarbitrary area, such as a particular parcel of land, a state, a country,the world, and so forth, with a district or square representing asub-area of the grid. Point data represents a value of a weathervariable at a particular point within the grid. The overarching goal isto fit a model that takes into account the information derived from bothdata sets. For instance, the areal averages provide an estimate for whatthe integral of the function representing the behavior of theenvironmental variable should be for a particular place and time,whereas the point values indicate the estimated value at particularlocations. Thus, the integral of the fitted model over a district of thegrid should approximate the estimate of the integral provided by theareal data while the behavior of the model should approximate the pointvalues for locations at or near the points for which values are providedby the point data.

In an embodiment, a hierarchical Bayesian state-space approach isutilized that decomposes generating blended estimates that vary in spaceand time into two distinct problems. The first problem addresseschange-of-support and statistical interpolation on the spatial side. Thesecond problem addresses how to account for seasonal and inter-annualvariability for various parameters on the temporal side.

Other features and aspect of the disclosure will become apparent in thedrawings, description, and claims.

2. Example Agricultural Intelligence Computer System

2.1 Structural Overview

FIG. 1 illustrates an example computer system that is configured toperform the functions described herein, shown in a field environmentwith other apparatus with which the system may interoperate. In oneembodiment, a user 102 owns, operates or possesses a field managercomputing device 104 in a field location or associated with a fieldlocation such as a field intended for agricultural activities or amanagement location for one or more agricultural fields. The fieldmanager computer device 104 is programmed or configured to provide fielddata 106 to an agricultural intelligence computer system 130 via one ormore networks 109.

Examples of field data 106 include (a) identification data (for example,acreage, field name, field identifiers, geographic identifiers, boundaryidentifiers, crop identifiers, and any other suitable data that may beused to identify farm land, such as a common land unit (CLU), lot andblock number, a parcel number, geographic coordinates and boundaries,Farm Serial Number (FSN), farm number, tract number, field number,section, township, and/or range), (b) harvest data (for example, croptype, crop variety, crop rotation, whether the crop is grownorganically, harvest date, Actual Production History (APH), expectedyield, yield, crop price, crop revenue, grain moisture, tillagepractice, and previous growing season information), (c) soil data (forexample, type, composition, pH, organic matter (OM), cation exchangecapacity (CEC)), (d) planting data (for example, planting date, seed(s)type, relative maturity (RM) of planted seed(s), seed population), (e)fertilizer data (for example, nutrient type (Nitrogen, Phosphorous,Potassium), application type, application date, amount, source, method),(f) pesticide data (for example, pesticide, herbicide, fungicide, othersubstance or mixture of substances intended for use as a plantregulator, defoliant, or desiccant, application date, amount, source,method), (g) irrigation data (for example, application date, amount,source, method), (h) weather data (for example, precipitation, rainfallrate, predicted rainfall, water runoff rate region, temperature, wind,forecast, pressure, visibility, clouds, heat index, dew point, humidity,snow depth, air quality, sunrise, sunset), (i) imagery data (forexample, imagery and light spectrum information from an agriculturalapparatus sensor, camera, computer, smartphone, tablet, unmanned aerialvehicle, planes or satellite), (j) scouting observations (photos,videos, free form notes, voice recordings, voice transcriptions, weatherconditions (temperature, precipitation (current and over time), soilmoisture, crop growth stage, wind velocity, relative humidity, dewpoint, black layer)), and (k) soil, seed, crop phenology, pest anddisease reporting, and predictions sources and databases.

A data server computer 108 is communicatively coupled to agriculturalintelligence computer system 130 and is programmed or configured to sendexternal data 110 to agricultural intelligence computer system 130 viathe network(s) 109. The external data server computer 108 may be ownedor operated by the same legal person or entity as the agriculturalintelligence computer system 130, or by a different person or entitysuch as a government agency, non-governmental organization (NGO), and/ora private data service provider. Examples of external data includeweather data, imagery data, soil data, or statistical data relating tocrop yields, among others. External data 110 may consist of the sametype of information as field data 106. In some embodiments, the externaldata 110 is provided by an external data server 108 owned by the sameentity that owns and/or operates the agricultural intelligence computersystem 130. For example, the agricultural intelligence computer system130 may include a data server focused exclusively on a type of data thatmight otherwise be obtained from third party sources, such as weatherdata. In some embodiments, an external data server 108 may actually beincorporated within the system 130.

An agricultural apparatus 111 may have one or more remote sensors 112fixed thereon, which sensors are communicatively coupled either directlyor indirectly via agricultural apparatus 111 to the agriculturalintelligence computer system 130 and are programmed or configured tosend sensor data to agricultural intelligence computer system 130.Examples of agricultural apparatus 111 include tractors, combines,harvesters, planters, trucks, fertilizer equipment, unmanned aerialvehicles, and any other item of physical machinery or hardware,typically mobile machinery, and which may be used in tasks associatedwith agriculture. In some embodiments, a single unit of apparatus 111may comprise a plurality of sensors 112 that are coupled locally in anetwork on the apparatus; controller area network (CAN) is example ofsuch a network that can be installed in combines or harvesters.Application controller 114 is communicatively coupled to agriculturalintelligence computer system 130 via the network(s) 109 and isprogrammed or configured to receive one or more scripts to control anoperating parameter of an agricultural vehicle or implement from theagricultural intelligence computer system 130. For instance, acontroller area network (CAN) bus interface may be used to enablecommunications from the agricultural intelligence computer system 130 tothe agricultural apparatus 111, such as how the CLIMATE FIELDVIEW DRIVE,available from The Climate Corporation, San Francisco, Calif., is used.Sensor data may consist of the same type of information as field data106. In some embodiments, remote sensors 112 may not be fixed to anagricultural apparatus 111 but may be remotely located in the field andmay communicate with network 109.

The apparatus 111 may comprise a cab computer 115 that is programmedwith a cab application, which may comprise a version or variant of themobile application for device 104 that is further described in othersections herein. In an embodiment, cab computer 115 comprises a compactcomputer, often a tablet-sized computer or smartphone, with a graphicalscreen display, such as a color display, that is mounted within anoperator's cab of the apparatus 111. Cab computer 115 may implement someor all of the operations and functions that are described further hereinfor the mobile computer device 104.

The network(s) 109 broadly represent any combination of one or more datacommunication networks including local area networks, wide areanetworks, internetworks or internets, using any of wireline or wirelesslinks, including terrestrial or satellite links. The network(s) may beimplemented by any medium or mechanism that provides for the exchange ofdata between the various elements of FIG. 1. The various elements ofFIG. 1 may also have direct (wired or wireless) communications links.The sensors 112, controller 114, external data server computer 108, andother elements of the system each comprise an interface compatible withthe network(s) 109 and are programmed or configured to use standardizedprotocols for communication across the networks such as TCP/IP,Bluetooth, CAN protocol and higher-layer protocols such as HTTP, TLS,and the like.

Agricultural intelligence computer system 130 is programmed orconfigured to receive field data 106 from field manager computing device104, external data 110 from external data server computer 108, andsensor data from remote sensor 112. Agricultural intelligence computersystem 130 may be further configured to host, use or execute one or morecomputer programs, other software elements, digitally programmed logicsuch as FPGAs or ASICs, or any combination thereof to performtranslation and storage of data values, construction of digital modelsof one or more crops on one or more fields, generation ofrecommendations and notifications, and generation and sending of scriptsto application controller 114, in the manner described further in othersections of this disclosure.

In an embodiment, agricultural intelligence computer system 130 isprogrammed with or comprises a communication layer 132, presentationlayer 134, data management layer 140, hardware/virtualization layer 150,and model and field data repository 160. “Layer,” in this context,refers to any combination of electronic digital interface circuits,microcontrollers, firmware such as drivers, and/or computer programs orother software elements.

Communication layer 132 may be programmed or configured to performinput/output interfacing functions including sending requests to fieldmanager computing device 104, external data server computer 108, andremote sensor 112 for field data, external data, and sensor datarespectively. Communication layer 132 may be programmed or configured tosend the received data to model and field data repository 160 to bestored as field data 106.

Presentation layer 134 may be programmed or configured to generate agraphical user interface (GUI) to be displayed on field managercomputing device 104, cab computer 115 or other computers that arecoupled to the system 130 through the network 109. The GUI may comprisecontrols for inputting data to be sent to agricultural intelligencecomputer system 130, generating requests for models and/orrecommendations, and/or displaying recommendations, notifications,models, and other field data.

Data management layer 140 may be programmed or configured to manage readoperations and write operations involving the repository 160 and otherfunctional elements of the system, including queries and result setscommunicated between the functional elements of the system and therepository. Examples of data management layer 140 include JDBC, SQLserver interface code, and/or HADOOP interface code, among others.Repository 160 may comprise a database. As used herein, the term“database” may refer to either a body of data, a relational databasemanagement system (RDBMS), or to both. As used herein, a database maycomprise any collection of data including hierarchical databases,relational databases, flat file databases, object-relational databases,object oriented databases, and any other structured collection ofrecords or data that is stored in a computer system. Examples of RDBMS'sinclude, but are not limited to including, ORACLE®, MYSQL, IBM® DB2,MICROSOFT® SQL SERVER, SYBASE®, and POSTGRESQL databases. However, anydatabase may be used that enables the systems and methods describedherein.

When field data 106 is not provided directly to the agriculturalintelligence computer system via one or more agricultural machines oragricultural machine devices that interacts with the agriculturalintelligence computer system, the user may be prompted via one or moreuser interfaces on the user device (served by the agriculturalintelligence computer system) to input such information. In an exampleembodiment, the user may specify identification data by accessing a mapon the user device (served by the agricultural intelligence computersystem) and selecting specific CLUs that have been graphically shown onthe map. In an alternative embodiment, the user 102 may specifyidentification data by accessing a map on the user device (served by theagricultural intelligence computer system 130) and drawing boundaries ofthe field over the map. Such CLU selection or map drawings representgeographic identifiers. In alternative embodiments, the user may specifyidentification data by accessing field identification data (provided asshape files or in a similar format) from the U. S. Department ofAgriculture Farm Service Agency or other source via the user device andproviding such field identification data to the agriculturalintelligence computer system.

In an example embodiment, the agricultural intelligence computer system130 is programmed to generate and cause displaying a graphical userinterface comprising a data manager for data input. After one or morefields have been identified using the methods described above, the datamanager may provide one or more graphical user interface widgets whichwhen selected can identify changes to the field, soil, crops, tillage,or nutrient practices. The data manager may include a timeline view, aspreadsheet view, and/or one or more editable programs.

FIG. 8 depicts an example embodiment of a timeline view for data entry.Using the display depicted in FIG. 8, a user computer can input aselection of a particular field and a particular date for the additionof event. Events depicted at the top of the timeline may includeNitrogen, Planting, Practices, and Soil. To add a nitrogen applicationevent, a user computer may provide input to select the nitrogen tab. Theuser computer may then select a location on the timeline for aparticular field in order to indicate an application of nitrogen on theselected field. In response to receiving a selection of a location onthe timeline for a particular field, the data manager may display a dataentry overlay, allowing the user computer to input data pertaining tonitrogen applications, planting procedures, soil application, tillageprocedures, irrigation practices, or other information relating to theparticular field. For example, if a user computer selects a portion ofthe timeline and indicates an application of nitrogen, then the dataentry overlay may include fields for inputting an amount of nitrogenapplied, a date of application, a type of fertilizer used, and any otherinformation related to the application of nitrogen.

In an embodiment, the data manager provides an interface for creatingone or more programs. “Program,” in this context, refers to a set ofdata pertaining to nitrogen applications, planting procedures, soilapplication, tillage procedures, irrigation practices, or otherinformation that may be related to one or more fields, and that can bestored in digital data storage for reuse as a set in other operations.After a program has been created, it may be conceptually applied to oneor more fields and references to the program may be stored in digitalstorage in association with data identifying the fields. Thus, insteadof manually entering identical data relating to the same nitrogenapplications for multiple different fields, a user computer may create aprogram that indicates a particular application of nitrogen and thenapply the program to multiple different fields. For example, in thetimeline view of FIG. 8, the top two timelines have the “Fall applied”program selected, which includes an application of 150 lbs. N/ac inearly April. The data manager may provide an interface for editing aprogram. In an embodiment, when a particular program is edited, eachfield that has selected the particular program is edited. For example,in FIG. 8, if the “Fall applied” program is edited to reduce theapplication of nitrogen to 130 lbs N/ac, the top two fields may beupdated with a reduced application of nitrogen based on the editedprogram.

In an embodiment, in response to receiving edits to a field that has aprogram selected, the data manager removes the correspondence of thefield to the selected program. For example, if a nitrogen application isadded to the top field in FIG. 8, the interface may update to indicatethat the “Fall applied” program is no longer being applied to the topfield. While the nitrogen application in early April may remain, updatesto the “Fall applied” program would not alter the April application ofnitrogen.

FIG. 9 depicts an example embodiment of a spreadsheet view for dataentry. Using the display depicted in FIG. 9, a user can create and editinformation for one or more fields. The data manager may includespreadsheets for inputting information with respect to Nitrogen,Planting, Practices, and Soil as depicted in FIG. 9. To edit aparticular entry, a user computer may select the particular entry in thespreadsheet and update the values. For example, FIG. 9 depicts anin-progress update to a target yield value for the second field.Additionally, a user computer may select one or more fields in order toapply one or more programs. In response to receiving a selection of aprogram for a particular field, the data manager may automaticallycomplete the entries for the particular field based on the selectedprogram. As with the timeline view, the data manager may update theentries for each field associated with a particular program in responseto receiving an update to the program. Additionally, the data managermay remove the correspondence of the selected program to the field inresponse to receiving an edit to one of the entries for the field.

In an embodiment, model and field data is stored in model and field datarepository 160. Model data comprises data models created for one or morefields. For example, a crop model may include a digitally constructedmodel of the development of a crop on the one or more fields. “Model,”in this context, refers to an electronic digitally stored set ofexecutable instructions and data values, associated with one another,which are capable of receiving and responding to a programmatic or otherdigital call, invocation, or request for resolution based upon specifiedinput values, to yield one or more stored output values that can serveas the basis of computer-implemented recommendations, output datadisplays, or machine control, among other things. Persons of skill inthe field find it convenient to express models using mathematicalequations, but that form of expression does not confine the modelsdisclosed herein to abstract concepts; instead, each model herein has apractical application in a computer in the form of stored executableinstructions and data that implement the model using the computer. Themodel data may include a model of past events on the one or more fields,a model of the current status of the one or more fields, and/or a modelof predicted events on the one or more fields. Model and field data maybe stored in data structures in memory, rows in a database table, inflat files or spreadsheets, or other forms of stored digital data.

Hardware/virtualization layer 150 comprises one or more centralprocessing units (CPUs), memory controllers, and other devices,components, or elements of a computer system such as volatile ornon-volatile memory, non-volatile storage such as disk, and I/O devicesor interfaces as illustrated and described, for example, in connectionwith FIG. 4. The layer 150 also may comprise programmed instructionsthat are configured to support virtualization, containerization, orother technologies.

For purposes of illustrating a clear example, FIG. 1 shows a limitednumber of instances of certain functional elements. However, in otherembodiments, there may be any number of such elements. For example,embodiments may use thousands or millions of different mobile computingdevices 104 associated with different users. Further, the system 130and/or external data server computer 108 may be implemented using two ormore processors, cores, clusters, or instances of physical machines orvirtual machines, configured in a discrete location or co-located withother elements in a datacenter, shared computing facility or cloudcomputing facility.

2.2. Application Program Overview

In an embodiment, the implementation of the functions described hereinusing one or more computer programs or other software elements that areloaded into and executed using one or more general-purpose computerswill cause the general-purpose computers to be configured as aparticular machine or as a computer that is specially adapted to performthe functions described herein. Further, each of the flow diagrams thatare described further herein may serve, alone or in combination with thedescriptions of processes and functions in prose herein, as algorithms,plans or directions that may be used to program a computer or logic toimplement the functions that are described. In other words, all theprose text herein, and all the drawing figures, together are intended toprovide disclosure of algorithms, plans or directions that aresufficient to permit a skilled person to program a computer to performthe functions that are described herein, in combination with the skilland knowledge of such a person given the level of skill that isappropriate for inventions and disclosures of this type.

In an embodiment, user 102 interacts with agricultural intelligencecomputer system 130 using field manager computing device 104 configuredwith an operating system and one or more application programs or apps;the field manager computing device 104 also may interoperate with theagricultural intelligence computer system independently andautomatically under program control or logical control and direct userinteraction is not always required. Field manager computing device 104broadly represents one or more of a smart phone, PDA, tablet computingdevice, laptop computer, desktop computer, workstation, or any othercomputing device capable of transmitting and receiving information andperforming the functions described herein. Field manager computingdevice 104 may communicate via a network using a mobile applicationstored on field manager computing device 104, and in some embodiments,the device may be coupled using a cable 113 or connector to the sensor112 and/or controller 114. A particular user 102 may own, operate orpossess and use, in connection with system 130, more than one fieldmanager computing device 104 at a time.

The mobile application may provide client-side functionality, via thenetwork to one or more mobile computing devices. In an exampleembodiment, field manager computing device 104 may access the mobileapplication via a web browser or a local client application or app.Field manager computing device 104 may transmit data to, and receivedata from, one or more front-end servers, using web-based protocols orformats such as HTTP, XML and/or JSON, or app-specific protocols. In anexample embodiment, the data may take the form of requests and userinformation input, such as field data, into the mobile computing device.In some embodiments, the mobile application interacts with locationtracking hardware and software on field manager computing device 104which determines the location of field manager computing device 104using standard tracking techniques such as multilateration of radiosignals, the global positioning system (GPS), WiFi positioning systems,or other methods of mobile positioning. In some cases, location data orother data associated with the device 104, user 102, and/or useraccount(s) may be obtained by queries to an operating system of thedevice or by requesting an app on the device to obtain data from theoperating system.

In an embodiment, field manager computing device 104 sends field data106 to agricultural intelligence computer system 130 comprising orincluding, but not limited to, data values representing one or more of:a geographical location of the one or more fields, tillage informationfor the one or more fields, crops planted in the one or more fields, andsoil data extracted from the one or more fields. Field manager computingdevice 104 may send field data 106 in response to user input from user102 specifying the data values for the one or more fields. Additionally,field manager computing device 104 may automatically send field data 106when one or more of the data values becomes available to field managercomputing device 104. For example, field manager computing device 104may be communicatively coupled to remote sensor 112 and/or applicationcontroller 114. In response to receiving data indicating thatapplication controller 114 released water onto the one or more fields,field manager computing device 104 may send field data 106 toagricultural intelligence computer system 130 indicating that water wasreleased on the one or more fields. Field data 106 identified in thisdisclosure may be input and communicated using electronic digital datathat is communicated between computing devices using parameterized URLsover HTTP, or another suitable communication or messaging protocol.

A commercial example of the mobile application is CLIMATE FIELDVIEW,commercially available from The Climate Corporation, San Francisco,Calif. The CLIMATE FIELDVIEW application, or other applications, may bemodified, extended, or adapted to include features, functions, andprogramming that have not been disclosed earlier than the filing date ofthis disclosure. In one embodiment, the mobile application comprises anintegrated software platform that allows a grower to make fact-baseddecisions for their operation because it combines historical data aboutthe grower's fields with any other data that the grower wishes tocompare. The combinations and comparisons may be performed in real timeand are based upon scientific models that provide potential scenarios topermit the grower to make better, more informed decisions.

FIG. 2 illustrates two views of an example logical organization of setsof instructions in main memory when an example mobile application isloaded for execution. In FIG. 2, each named element represents a regionof one or more pages of RAM or other main memory, or one or more blocksof disk storage or other non-volatile storage, and the programmedinstructions within those regions. In one embodiment, in view (a), amobile computer application 200 comprises account-fields-dataingestion-sharing instructions 202, overview and alert instructions 204,digital map book instructions 206, seeds and planting instructions 208,nitrogen instructions 210, weather instructions 212, field healthinstructions 214, and performance instructions 216.

In one embodiment, a mobile computer application 200 comprisesaccount-fields-data ingestion-sharing instructions 202 which areprogrammed to receive, translate, and ingest field data from third partysystems via manual upload or APIs. Data types may include fieldboundaries, yield maps, as-planted maps, soil test results, as-appliedmaps, and/or management zones, among others. Data formats may includeshape files, native data formats of third parties, and/or farmmanagement information system (FMIS) exports, among others. Receivingdata may occur via manual upload, e-mail with attachment, external APIsthat push data to the mobile application, or instructions that call APIsof external systems to pull data into the mobile application. In oneembodiment, mobile computer application 200 comprises a data inbox. Inresponse to receiving a selection of the data inbox, the mobile computerapplication 200 may display a graphical user interface for manuallyuploading data files and importing uploaded files to a data manager.

In one embodiment, digital map book instructions 206 comprise field mapdata layers stored in device memory and are programmed with datavisualization tools and geospatial field notes. This provides growerswith convenient information close at hand for reference, logging andvisual insights into field performance. In one embodiment, overview andalert instructions 204 are programmed to provide an operation-wide viewof what is important to the grower, and timely recommendations to takeaction or focus on particular issues. This permits the grower to focustime on what needs attention, to save time and preserve yield throughoutthe season. In one embodiment, seeds and planting instructions 208 areprogrammed to provide tools for seed selection, hybrid placement, andscript creation, including variable rate (VR) script creation, basedupon scientific models and empirical data. This enables growers tomaximize yield or return on investment through optimized seed purchase,placement and population.

In one embodiment, script generation instructions 205 are programmed toprovide an interface for generating scripts, including variable rate(VR) fertility scripts. The interface enables growers to create scriptsfor field implements, such as nutrient applications, planting, andirrigation. For example, a planting script interface may comprise toolsfor identifying a type of seed for planting. Upon receiving a selectionof the seed type, mobile computer application 200 may display one ormore fields broken into management zones, such as the field map datalayers created as part of digital map book instructions 206. In oneembodiment, the management zones comprise soil zones along with a panelidentifying each soil zone and a soil name, texture, drainage for eachzone, or other field data. Mobile computer application 200 may alsodisplay tools for editing or creating such, such as graphical tools fordrawing management zones, such as soil zones, over a map of one or morefields. Planting procedures may be applied to all management zones ordifferent planting procedures may be applied to different subsets ofmanagement zones. When a script is created, mobile computer application200 may make the script available for download in a format readable byan application controller, such as an archived or compressed format.Additionally, and/or alternatively, a script may be sent directly to cabcomputer 115 from mobile computer application 200 and/or uploaded to oneor more data servers and stored for further use. In one embodiment,nitrogen instructions 210 are programmed to provide tools to informnitrogen decisions by visualizing the availability of nitrogen to crops.This enables growers to maximize yield or return on investment throughoptimized nitrogen application during the season. Example programmedfunctions include displaying images such as SSURGO images to enabledrawing of application zones and/or images generated from subfield soildata, such as data obtained from sensors, at a high spatial resolution(as fine as 10 meters or smaller because of their proximity to thesoil); upload of existing grower-defined zones; providing an applicationgraph and/or a map to enable tuning application(s) of nitrogen acrossmultiple zones; output of scripts to drive machinery; tools for massdata entry and adjustment; and/or maps for data visualization, amongothers. “Mass data entry,” in this context, may mean entering data onceand then applying the same data to multiple fields that have beendefined in the system; example data may include nitrogen applicationdata that is the same for many fields of the same grower, but such massdata entry applies to the entry of any type of field data into themobile computer application 200. For example, nitrogen instructions 210may be programmed to accept definitions of nitrogen planting andpractices programs and to accept user input specifying to apply thoseprograms across multiple fields. “Nitrogen planting programs,” in thiscontext, refers to a stored, named set of data that associates: a name,color code or other identifier, one or more dates of application, typesof material or product for each of the dates and amounts, method ofapplication or incorporation such as injected or knifed in, and/oramounts or rates of application for each of the dates, crop or hybridthat is the subject of the application, among others. “Nitrogenpractices programs,” in this context, refers to a stored, named set ofdata that associates: a practices name; a previous crop; a tillagesystem; a date of primarily tillage; one or more previous tillagesystems that were used; one or more indicators of application type, suchas manure, that were used. Nitrogen instructions 210 also may beprogrammed to generate and cause displaying a nitrogen graph, whichindicates projections of plant use of the specified nitrogen and whethera surplus or shortfall is predicted; in some embodiments, differentcolor indicators may signal a magnitude of surplus or magnitude ofshortfall. In one embodiment, a nitrogen graph comprises a graphicaldisplay in a computer display device comprising a plurality of rows,each row associated with and identifying a field; data specifying whatcrop is planted in the field, the field size, the field location, and agraphic representation of the field perimeter; in each row, a timelineby month with graphic indicators specifying each nitrogen applicationand amount at points correlated to month names; and numeric and/orcolored indicators of surplus or shortfall, in which color indicatesmagnitude.

In one embodiment, the nitrogen graph may include one or more user inputfeatures, such as dials or slider bars, to dynamically change thenitrogen planting and practices programs so that a user may optimize hisnitrogen graph. The user may then use his optimized nitrogen graph andthe related nitrogen planting and practices programs to implement one ormore scripts, including variable rate (VR) fertility scripts. Nitrogeninstructions 210 also may be programmed to generate and cause displayinga nitrogen map, which indicates projections of plant use of thespecified nitrogen and whether a surplus or shortfall is predicted; insome embodiments, different color indicators may signal a magnitude ofsurplus or magnitude of shortfall. The nitrogen map may displayprojections of plant use of the specified nitrogen and whether a surplusor shortfall is predicted for different times in the past and the future(such as daily, weekly, monthly or yearly) using numeric and/or coloredindicators of surplus or shortfall, in which color indicates magnitude.In one embodiment, the nitrogen map may include one or more user inputfeatures, such as dials or slider bars, to dynamically change thenitrogen planting and practices programs so that a user may optimize hisnitrogen map, such as to obtain a preferred amount of surplus toshortfall. The user may then use his optimized nitrogen map and therelated nitrogen planting and practices programs to implement one ormore scripts, including variable rate (VR) fertility scripts. In otherembodiments, similar instructions to the nitrogen instructions 210 couldbe used for application of other nutrients (such as phosphorus andpotassium) application of pesticide, and irrigation programs.

In one embodiment, weather instructions 212 are programmed to providefield-specific recent weather data and forecasted weather information.This enables growers to save time and have an efficient integrateddisplay with respect to daily operational decisions.

In one embodiment, field health instructions 214 are programmed toprovide timely remote sensing images highlighting in-season cropvariation and potential concerns. Example programmed functions includecloud checking, to identify possible clouds or cloud shadows;determining nitrogen indices based on field images; graphicalvisualization of scouting layers, including, for example, those relatedto field health, and viewing and/or sharing of scouting notes; and/ordownloading satellite images from multiple sources and prioritizing theimages for the grower, among others.

In one embodiment, performance instructions 216 are programmed toprovide reports, analysis, and insight tools using on-farm data forevaluation, insights and decisions. This enables the grower to seekimproved outcomes for the next year through fact-based conclusions aboutwhy return on investment was at prior levels, and insight intoyield-limiting factors. The performance instructions 216 may beprogrammed to communicate via the network(s) 109 to back-end analyticsprograms executed at agricultural intelligence computer system 130and/or external data server computer 108 and configured to analyzemetrics such as yield, hybrid, population, SSURGO, soil tests, orelevation, among others. Programmed reports and analysis may includeyield variability analysis, benchmarking of yield and other metricsagainst other growers based on anonymized data collected from manygrowers, or data for seeds and planting, among others.

Applications having instructions configured in this way may beimplemented for different computing device platforms while retaining thesame general user interface appearance. For example, the mobileapplication may be programmed for execution on tablets, smartphones, orserver computers that are accessed using browsers at client computers.Further, the mobile application as configured for tablet computers orsmartphones may provide a full app experience or a cab app experiencethat is suitable for the display and processing capabilities of cabcomputer 115. For example, referring now to view (b) of FIG. 2, in oneembodiment a cab computer application 220 may comprise maps-cabinstructions 222, remote view instructions 224, data collect andtransfer instructions 226, machine alerts instructions 228, scripttransfer instructions 230, and scouting-cab instructions 232. The codebase for the instructions of view (b) may be the same as for view (a)and executables implementing the code may be programmed to detect thetype of platform on which they are executing and to expose, through agraphical user interface, only those functions that are appropriate to acab platform or full platform. This approach enables the system torecognize the distinctly different user experience that is appropriatefor an in-cab environment and the different technology environment ofthe cab. The maps-cab instructions 222 may be programmed to provide mapviews of fields, farms or regions that are useful in directing machineoperation. The remote view instructions 224 may be programmed to turnon, manage, and provide views of machine activity in real-time or nearreal-time to other computing devices connected to the system 130 viawireless networks, wired connectors or adapters, and the like. The datacollect and transfer instructions 226 may be programmed to turn on,manage, and provide transfer of data collected at machine sensors andcontrollers to the system 130 via wireless networks, wired connectors oradapters, and the like. The machine alerts instructions 228 may beprogrammed to detect issues with operations of the machine or tools thatare associated with the cab and generate operator alerts. The scripttransfer instructions 230 may be configured to transfer in scripts ofinstructions that are configured to direct machine operations or thecollection of data. The scouting-cab instructions 230 may be programmedto display location-based alerts and information received from thesystem 130 based on the location of the agricultural apparatus 111 orsensors 112 in the field and ingest, manage, and provide transfer oflocation-based scouting observations to the system 130 based on thelocation of the agricultural apparatus 111 or sensors 112 in the field.

2.3. Data Ingest to the Computer System

In an embodiment, external data server computer 108 stores external data110, including soil data representing soil composition for the one ormore fields and weather data representing temperature and precipitationon the one or more fields. The weather data may include past and presentweather data as well as forecasts for future weather data. In anembodiment, external data server computer 108 comprises a plurality ofservers hosted by different entities. For example, a first server maycontain soil composition data while a second server may include weatherdata. Additionally, soil composition data may be stored in multipleservers. For example, one server may store data representing percentageof sand, silt, and clay in the soil while a second server may store datarepresenting percentage of organic matter (OM) in the soil.

In an embodiment, remote sensor 112 comprises one or more sensors thatare programmed or configured to produce one or more observations. Remotesensor 112 may be aerial sensors, such as satellites, vehicle sensors,planting equipment sensors, tillage sensors, fertilizer or insecticideapplication sensors, harvester sensors, and any other implement capableof receiving data from the one or more fields. In an embodiment,application controller 114 is programmed or configured to receiveinstructions from agricultural intelligence computer system 130.Application controller 114 may also be programmed or configured tocontrol an operating parameter of an agricultural vehicle or implement.For example, an application controller may be programmed or configuredto control an operating parameter of a vehicle, such as a tractor,planting equipment, tillage equipment, fertilizer or insecticideequipment, harvester equipment, or other farm implements such as a watervalve. Other embodiments may use any combination of sensors andcontrollers, of which the following are merely selected examples.

The system 130 may obtain or ingest data under user 102 control, on amass basis from a large number of growers who have contributed data to ashared database system. This form of obtaining data may be termed“manual data ingest” as one or more user-controlled computer operationsare requested or triggered to obtain data for use by the system 130. Asan example, the CLIMATE FIELDVIEW application, commercially availablefrom The Climate Corporation, San Francisco, Calif., may be operated toexport data to system 130 for storing in the repository 160.

For example, seed monitor systems can both control planter apparatuscomponents and obtain planting data, including signals from seed sensorsvia a signal harness that comprises a CAN backbone and point-to-pointconnections for registration and/or diagnostics. Seed monitor systemscan be programmed or configured to display seed spacing, population andother information to the user via the cab computer 115 or other deviceswithin the system 130. Examples are disclosed in U.S. Pat. No. 8,738,243and US Pat. Pub. 20150094916, and the present disclosure assumesknowledge of those other patent disclosures.

Likewise, yield monitor systems may contain yield sensors for harvesterapparatus that send yield measurement data to the cab computer 115 orother devices within the system 130. Yield monitor systems may utilizeone or more remote sensors 112 to obtain grain moisture measurements ina combine or other harvester and transmit these measurements to the uservia the cab computer 115 or other devices within the system 130.

In an embodiment, examples of sensors 112 that may be used with anymoving vehicle or apparatus of the type described elsewhere hereininclude kinematic sensors and position sensors. Kinematic sensors maycomprise any of speed sensors such as radar or wheel speed sensors,accelerometers, or gyros. Position sensors may comprise GPS receivers ortransceivers, or WiFi-based position or mapping apps that are programmedto determine location based upon nearby WiFi hotspots, among others.

In an embodiment, examples of sensors 112 that may be used with tractorsor other moving vehicles include engine speed sensors, fuel consumptionsensors, area counters or distance counters that interact with GPS orradar signals, PTO (power take-off) speed sensors, tractor hydraulicssensors configured to detect hydraulics parameters such as pressure orflow, and/or and hydraulic pump speed, wheel speed sensors or wheelslippage sensors. In an embodiment, examples of controllers 114 that maybe used with tractors include hydraulic directional controllers,pressure controllers, and/or flow controllers; hydraulic pump speedcontrollers; speed controllers or governors; hitch position controllers;or wheel position controllers provide automatic steering.

In an embodiment, examples of sensors 112 that may be used with seedplanting equipment such as planters, drills, or air seeders include seedsensors, which may be optical, electromagnetic, or impact sensors;downforce sensors such as load pins, load cells, pressure sensors; soilproperty sensors such as reflectivity sensors, moisture sensors,electrical conductivity sensors, optical residue sensors, or temperaturesensors; component operating criteria sensors such as planting depthsensors, downforce cylinder pressure sensors, seed disc speed sensors,seed drive motor encoders, seed conveyor system speed sensors, or vacuumlevel sensors; or pesticide application sensors such as optical or otherelectromagnetic sensors, or impact sensors. In an embodiment, examplesof controllers 114 that may be used with such seed planting equipmentinclude: toolbar fold controllers, such as controllers for valvesassociated with hydraulic cylinders; downforce controllers, such ascontrollers for valves associated with pneumatic cylinders, airbags, orhydraulic cylinders, and programmed for applying downforce to individualrow units or an entire planter frame; planting depth controllers, suchas linear actuators; metering controllers, such as electric seed meterdrive motors, hydraulic seed meter drive motors, or swath controlclutches; hybrid selection controllers, such as seed meter drive motors,or other actuators programmed for selectively allowing or preventingseed or an air-seed mixture from delivering seed to or from seed metersor central bulk hoppers; metering controllers, such as electric seedmeter drive motors, or hydraulic seed meter drive motors; seed conveyorsystem controllers, such as controllers for a belt seed deliveryconveyor motor; marker controllers, such as a controller for a pneumaticor hydraulic actuator; or pesticide application rate controllers, suchas metering drive controllers, orifice size or position controllers.

In an embodiment, examples of sensors 112 that may be used with tillageequipment include position sensors for tools such as shanks or discs;tool position sensors for such tools that are configured to detectdepth, gang angle, or lateral spacing; downforce sensors; or draft forcesensors. In an embodiment, examples of controllers 114 that may be usedwith tillage equipment include downforce controllers or tool positioncontrollers, such as controllers configured to control tool depth, gangangle, or lateral spacing.

In an embodiment, examples of sensors 112 that may be used in relationto apparatus for applying fertilizer, insecticide, fungicide and thelike, such as on-planter starter fertilizer systems, subsoil fertilizerapplicators, or fertilizer sprayers, include: fluid system criteriasensors, such as flow sensors or pressure sensors; sensors indicatingwhich spray head valves or fluid line valves are open; sensorsassociated with tanks, such as fill level sensors; sectional orsystem-wide supply line sensors, or row-specific supply line sensors; orkinematic sensors such as accelerometers disposed on sprayer booms. Inan embodiment, examples of controllers 114 that may be used with suchapparatus include pump speed controllers; valve controllers that areprogrammed to control pressure, flow, direction, PWM and the like; orposition actuators, such as for boom height, subsoiler depth, or boomposition.

In an embodiment, examples of sensors 112 that may be used withharvesters include yield monitors, such as impact plate strain gauges orposition sensors, capacitive flow sensors, load sensors, weight sensors,or torque sensors associated with elevators or augers, or optical orother electromagnetic grain height sensors; grain moisture sensors, suchas capacitive sensors; grain loss sensors, including impact, optical, orcapacitive sensors; header operating criteria sensors such as headerheight, header type, deck plate gap, feeder speed, and reel speedsensors; separator operating criteria sensors, such as concaveclearance, rotor speed, shoe clearance, or chaffer clearance sensors;auger sensors for position, operation, or speed; or engine speedsensors. In an embodiment, examples of controllers 114 that may be usedwith harvesters include header operating criteria controllers forelements such as header height, header type, deck plate gap, feederspeed, or reel speed; separator operating criteria controllers forfeatures such as concave clearance, rotor speed, shoe clearance, orchaffer clearance; or controllers for auger position, operation, orspeed.

In an embodiment, examples of sensors 112 that may be used with graincarts include weight sensors, or sensors for auger position, operation,or speed. In an embodiment, examples of controllers 114 that may be usedwith grain carts include controllers for auger position, operation, orspeed.

In an embodiment, examples of sensors 112 and controllers 114 may beinstalled in unmanned aerial vehicle (UAV) apparatus or “drones.” Suchsensors may include cameras with detectors effective for any range ofthe electromagnetic spectrum including visible light, infrared,ultraviolet, near-infrared (NIR), and the like; accelerometers;altimeters; temperature sensors; humidity sensors; pitot tube sensors orother airspeed or wind velocity sensors; battery life sensors; or radaremitters and reflected radar energy detection apparatus. Suchcontrollers may include guidance or motor control apparatus, controlsurface controllers, camera controllers, or controllers programmed toturn on, operate, obtain data from, manage and configure any of theforegoing sensors. Examples are disclosed in U.S. patent applicationSer. No. 14/831,165 and the present disclosure assumes knowledge of thatother patent disclosure.

In an embodiment, sensors 112 and controllers 114 may be affixed to soilsampling and measurement apparatus that is configured or programmed tosample soil and perform soil chemistry tests, soil moisture tests, andother tests pertaining to soil. For example, the apparatus disclosed inU.S. Pat. Nos. 8,767,194 and 8,712,148 may be used, and the presentdisclosure assumes knowledge of those patent disclosures.

In another embodiment, sensors 112 and controllers 114 may compriseweather devices for monitoring weather conditions of fields. Forexample, the apparatus disclosed in International Pat. Application No.PCT/US2016/029609 may be used, and the present disclosure assumesknowledge of those patent disclosures.

2.4 Process Overview-Agronomic Model Training

In an embodiment, the agricultural intelligence computer system 130 isprogrammed or configured to create an agronomic model. In this context,an agronomic model is a data structure in memory of the agriculturalintelligence computer system 130 that comprises field data 106, such asidentification data and harvest data for one or more fields. Theagronomic model may also comprise calculated agronomic properties whichdescribe either conditions which may affect the growth of one or morecrops on a field, or properties of the one or more crops, or both.Additionally, an agronomic model may comprise recommendations based onagronomic factors such as crop recommendations, irrigationrecommendations, planting recommendations, and harvestingrecommendations. The agronomic factors may also be used to estimate oneor more crop related results, such as agronomic yield. The agronomicyield of a crop is an estimate of quantity of the crop that is produced,or in some examples the revenue or profit obtained from the producedcrop.

In an embodiment, the agricultural intelligence computer system 130 mayuse a preconfigured agronomic model to calculate agronomic propertiesrelated to currently received location and crop information for one ormore fields. The preconfigured agronomic model is based upon previouslyprocessed field data, including but not limited to, identification data,harvest data, fertilizer data, and weather data. The preconfiguredagronomic model may have been cross validated to ensure accuracy of themodel. Cross validation may include comparison to ground truthing thatcompares predicted results with actual results on a field, such as acomparison of precipitation estimate with a rain gauge or sensorproviding weather data at the same or nearby location or an estimate ofnitrogen content with a soil sample measurement.

FIG. 3 illustrates a programmed process by which the agriculturalintelligence computer system generates one or more preconfiguredagronomic models using field data provided by one or more data sources.FIG. 3 may serve as an algorithm or instructions for programming thefunctional elements of the agricultural intelligence computer system 130to perform the operations that are now described.

At block 305, the agricultural intelligence computer system 130 isconfigured or programmed to implement agronomic data preprocessing offield data received from one or more data sources. The field datareceived from one or more data sources may be preprocessed for thepurpose of removing noise and distorting effects within the agronomicdata including measured outliers that would bias received field datavalues. Embodiments of agronomic data preprocessing may include, but arenot limited to, removing data values commonly associated with outlierdata values, specific measured data points that are known tounnecessarily skew other data values, data smoothing techniques used toremove or reduce additive or multiplicative effects from noise, andother filtering or data derivation techniques used to provide cleardistinctions between positive and negative data inputs.

At block 310, the agricultural intelligence computer system 130 isconfigured or programmed to perform data subset selection using thepreprocessed field data in order to identify datasets useful for initialagronomic model generation. The agricultural intelligence computersystem 130 may implement data subset selection techniques including, butnot limited to, a genetic algorithm method, an all subset models method,a sequential search method, a stepwise regression method, a particleswarm optimization method, and an ant colony optimization method. Forexample, a genetic algorithm selection technique uses an adaptiveheuristic search algorithm, based on evolutionary principles of naturalselection and genetics, to determine and evaluate datasets within thepreprocessed agronomic data.

At block 315, the agricultural intelligence computer system 130 isconfigured or programmed to implement field dataset evaluation. In anembodiment, a specific field dataset is evaluated by creating anagronomic model and using specific quality thresholds for the createdagronomic model. Agronomic models may be compared using cross validationtechniques including, but not limited to, root mean square error ofleave-one-out cross validation (RMSECV), mean absolute error, and meanpercentage error. For example, RMSECV can cross validate agronomicmodels by comparing predicted agronomic property values created by theagronomic model against historical agronomic property values collectedand analyzed. In an embodiment, the agronomic dataset evaluation logicis used as a feedback loop where agronomic datasets that do not meetconfigured quality thresholds are used during future data subsetselection steps (block 310).

At block 320, the agricultural intelligence computer system 130 isconfigured or programmed to implement agronomic model creation basedupon the cross validated agronomic datasets. In an embodiment, agronomicmodel creation may implement multivariate regression techniques tocreate preconfigured agronomic data models.

At block 325, the agricultural intelligence computer system 130 isconfigured or programmed to store the preconfigured agronomic datamodels for future field data evaluation.

2.5 Data Blending Subsystem

In an embodiment, data blending subsystem 170 includes components thatretrieve observations (including point measurements and areal averages),process the observations to fit a model that blends the pointmeasurements and areal averages, and then uses the fitted model togenerate predictions for previously unknown spatio-temporal coordinates.The data blending subsystem 170 and the components contain therein mayrepresent software instructions (for example source and/or compiledprogram code), hardware components (for example application-specificintegrated circuits and/or field programmable gate arrays), orcombinations thereof. In an embodiment, the data blending subsystem 170includes weather data input module 171, model fitting module 172, andweather analysis module 173.

2.5.1 Data Blending Subsystem Functional Overview

FIG. 5 illustrates an example functional overview for the data blendingsubsystem 170 according to an embodiment. In other embodiments, thedepicted modules may be divided out into further sub-modules or combinedinto modules that are responsible for a greater number of tasks thanthose depicted in FIG. 5.

In FIG. 5, the weather data input module 171 retrieves observation data501 for a particular environmental variable and passes the observationdata 501 to the model fitting module 172. The observation data 501 mayinclude point measurements (for example value and variance of theenvironmental variable at a particular point) and areal measurements(for example value and variance of the environmental variable averagedover a particular district of the grid). The observation data 501 may becollected from a variety of different sources, as is explained below infurther detail in Section 3.0 (“Example System Inputs”).

In an embodiment the model fitting module 172 constructs a state-spacemodel describing the behavior of the environmental variable over spaceand time and fits the model to the observation data 501, resulting infitted model 502. A detailed description of the model is described belowin Section 4.0 (“Gaussian Model”) and an example process for performingthe fitting is described below in Section 5.0 (“Data Blending ProcessFlow”). In an embodiment, the fitted model 502 includes the mean andvariances of a gridded set of latent Gaussian processes that, combinedwith an observation equation representing the harmonic frequencies ofthe environmental variable (for example yearly and seasonaloscillations), provides the behavior of the environmental variable overspace and time.

In an embodiment, the weather analysis module 173 receives the fittedmodel 502 and a time and location 504 representing a new spatio-temporalcoordinate. The weather analysis model 173 then uses the fitted model502 to generate a predicted value and variance 505 of the environmentalvariable at the new spatio-temporal coordinate specified by the time andlocation 504.

2.6 Implementation Example—Hardware Overview

According to one embodiment, the techniques described herein areimplemented by one or more special-purpose computing devices. Thespecial-purpose computing devices may be hard-wired to perform thetechniques, or may include digital electronic devices such as one ormore application-specific integrated circuits (ASICs) or fieldprogrammable gate arrays (FPGAs) that are persistently programmed toperform the techniques, or may include one or more general purposehardware processors programmed to perform the techniques pursuant toprogram instructions in firmware, memory, other storage, or acombination. Such special-purpose computing devices may also combinecustom hard-wired logic, ASICs, or FPGAs with custom programming toaccomplish the techniques. The special-purpose computing devices may bedesktop computer systems, portable computer systems, handheld devices,networking devices or any other device that incorporates hard-wiredand/or program logic to implement the techniques.

For example, FIG. 4 is a block diagram that illustrates a computersystem 400 upon which an embodiment of the invention may be implemented.Computer system 400 includes a bus 402 or other communication mechanismfor communicating information, and a hardware processor 404 coupled withbus 402 for processing information. Hardware processor 404 may be, forexample, a general-purpose microprocessor.

Computer system 400 also includes a main memory 406, such as arandom-access memory (RAM) or other dynamic storage device, coupled tobus 402 for storing information and instructions to be executed byprocessor 404. Main memory 406 also may be used for storing temporaryvariables or other intermediate information during execution ofinstructions to be executed by processor 404. Such instructions, whenstored in non-transitory storage media accessible to processor 404,render computer system 400 into a special-purpose machine that iscustomized to perform the operations specified in the instructions.

Computer system 400 further includes a read only memory (ROM) 408 orother static storage device coupled to bus 402 for storing staticinformation and instructions for processor 404. A storage device 410,such as a magnetic disk, optical disk, or solid-state drive is providedand coupled to bus 402 for storing information and instructions.

Computer system 400 may be coupled via bus 402 to a display 412, such asa cathode ray tube (CRT), for displaying information to a computer user.An input device 414, including alphanumeric and other keys, is coupledto bus 402 for communicating information and command selections toprocessor 404. Another type of user input device is cursor control 416,such as a mouse, a trackball, or cursor direction keys for communicatingdirection information and command selections to processor 404 and forcontrolling cursor movement on display 412. This input device typicallyhas two degrees of freedom in two axes, a first axis (e.g., x) and asecond axis (e.g., y), that allows the device to specify positions in aplane.

Computer system 400 may implement the techniques described herein usingcustomized hard-wired logic, one or more ASICs or FPGAs, firmware and/orprogram logic which in combination with the computer system causes orprograms computer system 400 to be a special-purpose machine. Accordingto one embodiment, the techniques herein are performed by computersystem 400 in response to processor 404 executing one or more sequencesof one or more instructions contained in main memory 406. Suchinstructions may be read into main memory 406 from another storagemedium, such as storage device 410. Execution of the sequences ofinstructions contained in main memory 406 causes processor 404 toperform the process steps described herein. In alternative embodiments,hard-wired circuitry may be used in place of or in combination withsoftware instructions.

The term “storage media” as used herein refers to any non-transitorymedia that store data and/or instructions that cause a machine tooperate in a specific fashion. Such storage media may comprisenon-volatile media and/or volatile media. Non-volatile media includes,for example, optical disks, magnetic disks, or solid-state drives, suchas storage device 410. Volatile media includes dynamic memory, such asmain memory 406. Common forms of storage media include, for example, afloppy disk, a flexible disk, hard disk, solid-state drive, magnetictape, or any other magnetic data storage medium, a CD-ROM, any otheroptical data storage medium, any physical medium with patterns of holes,a RAM, a PROM, and EPROM, a FLASH-EPROM, NVRAM, any other memory chip orcartridge.

Storage media is distinct from but may be used in conjunction withtransmission media. Transmission media participates in transferringinformation between storage media. For example, transmission mediaincludes coaxial cables, copper wire and fiber optics, including thewires that comprise bus 402. Transmission media can also take the formof acoustic or light waves, such as those generated during radio-waveand infra-red data communications.

Various forms of media may be involved in carrying one or more sequencesof one or more instructions to processor 404 for execution. For example,the instructions may initially be carried on a magnetic disk orsolid-state drive of a remote computer. The remote computer can load theinstructions into its dynamic memory and send the instructions over atelephone line using a modem. A modem local to computer system 400 canreceive the data on the telephone line and use an infra-red transmitterto convert the data to an infra-red signal. An infra-red detector canreceive the data carried in the infra-red signal and appropriatecircuitry can place the data on bus 402. Bus 402 carries the data tomain memory 406, from which processor 404 retrieves and executes theinstructions. The instructions received by main memory 406 mayoptionally be stored on storage device 410 either before or afterexecution by processor 404.

Computer system 400 also includes a communication interface 418 coupledto bus 402. Communication interface 418 provides a two-way datacommunication coupling to a network link 420 that is connected to alocal network 422. For example, communication interface 418 may be anintegrated services digital network (ISDN) card, cable modem, satellitemodem, or a modem to provide a data communication connection to acorresponding type of telephone line. As another example, communicationinterface 418 may be a local area network (LAN) card to provide a datacommunication connection to a compatible LAN. Wireless links may also beimplemented. In any such implementation, communication interface 418sends and receives electrical, electromagnetic or optical signals thatcarry digital data streams representing various types of information.

Network link 420 typically provides data communication through one ormore networks to other data devices. For example, network link 420 mayprovide a connection through local network 422 to a host computer 424 orto data equipment operated by an Internet Service Provider (ISP) 426.ISP 426 in turn provides data communication services through theworld-wide packet data communication network now commonly referred to asthe “Internet” 428. Local network 422 and Internet 428 both useelectrical, electromagnetic or optical signals that carry digital datastreams. The signals through the various networks and the signals onnetwork link 420 and through communication interface 418, which carrythe digital data to and from computer system 400, are example forms oftransmission media.

Computer system 400 can send messages and receive data, includingprogram code, through the network(s), network link 420 and communicationinterface 418. In the Internet example, a server 430 might transmit arequested code for an application program through Internet 428, ISP 426,local network 422 and communication interface 418.

The received code may be executed by processor 404 as it is received,and/or stored in storage device 410, or other non-volatile storage forlater execution.

3.0 Example System Inputs

In an embodiment, the observation data 501 that the data blendingsubsystem 170 uses to generate the fitted model 502 contains both pointdata and areal averages. Point data represents values and variances of aspecific environmental variable that are recorded at variousgeographical coordinates. For example, one common source for point dataare weather stations or sensors which are spread (often non-uniformly)across the area under observation. For instance, NOAA METAR station dataavailable from the Aviation Weather Center could be used for the pointdata. Areal averages represent values and variances of a specificenvironmental variable that have been averaged across each district of agrid superimposed over the area under observation. For example, onecommon source for areal averages are analyses and reanalyses which havebeen constructed by simulating the physical processes which govern thebehavior of the environmental variable over time from an initialstarting state. For instance, areal averages may be obtained from theECMWF ERA-Interim analysis product available from the European Centerfor Medium-Range Weather Forecasts. Another common source of arealaverages include remote sensing data, such as pixel data collected fromsatellites, where the “grid” is dependent on the spatial resolution atwhich the satellite or other remote sensing device can takemeasurements.

In some embodiments, in order to obtain the observation data 501, thedata blending subsystem 170, via the weather data input module 171,accesses external data 110 stored on servers/databases belonging toweather statistics reporting agencies (such as the Aviation WeatherCenter and European Center for Medium-Range Weather Forecasts) bysending requests in any number of formats, such as HTTP requests, FTPrequests, and so forth. In some embodiments, the weather data inputmodule 171 proactively obtains the observation data 501 and stores theobservation data in the model data and field data repository 160 forlocal processing by the other components of the data blending subsystem170. For example, the weather data input module 171 may send periodicrequests for updates to the external data 110 and then cache thereturned information in the model data and field data repository 160 forlocal processing. In other embodiments, the weather data input module171 obtains the external data 110 in response to receiving a requestfrom the field manager computing device 104 or an application executingon the agricultural intelligence computer system 130 or an externalclient which invokes the data blending subsystem 170. The request mayspecify from which sources the observation data 501 should be obtainedand/or the environmental variable for which the observation data 501should be obtained.

4.0 Gaussian Model

4.1 Notation

Throughout the following description the following notation is used:unknown quantities (for example, stochastic processes and time-invariantparameters) are represented by Greek letters; observables and fixedquantities are denoted by upper case Roman letters (for example, Y);lower case Roman letters are employed primarily as indexes for arrays,and as auxiliary variables; matrices and vectors are typed in uppercaseand lowercase bold font, respectively (for example, K and θ); tilde isread as “follows the distribution” (for example Y˜Normal[0,1]); and theprime character ′ denotes transpose (for example, α′).

4.2 Temporal Variability

The primary concern with respect to temporal variability whencharacterizing the hierarchical dynamic model is to decompose the timevarying information present in the observation data 501.

Consider a set of N observations made on day t, t∈{1, 2, . . . , T}, atspatial locations s₁, . . . , s_(N), within a two-dimensional domain.Let Y_(t) ^(raw)(s_(i)) denote the random variable associated with thei-th raw observation. The set of observations can be viewed as acombination of oscillations with different frequencies (annual,semi-annual, quarterly, and so forth), a time-varying baseline, andGaussian white noise (Equation 1.0):Y _(t) ^(raw)(s _(i))=x _(t)(s _(i))+ζ_(t)(s _(i)),Y _(t) ^(raw)(s _(i))=x _(t)(s _(i))+ζ_(t)(s _(i)),where (Equation 2.0):

${{x_{t}( s_{i} )} = {{a_{1,t}( s_{i} )} + {\sum\limits_{f = 1}^{L}{b_{f} \times \{ {{{a_{{2f},t}( s_{i} )}{\sin\lbrack \frac{2\pi\mspace{14mu}{ft}}{365} \rbrack}} + {{a_{{{2f} + 1},t}( s_{i} )}{\cos\lbrack \frac{2\pi\mspace{14mu}{ft}}{365} \rbrack}}} \}}}}},\mspace{20mu}{and}$$\mspace{20mu}{{\zeta_{t}( s_{i} )}\text{∼}{{Normal}\mspace{11mu}\lbrack {0,\frac{v^{2}}{\tau}} \rbrack}}$is noise with variance v²/τ. Equation 2.0 above assumes that leap days(February 29) are disregarded, leaving 1/365 day⁻¹ as corresponding tothe annual fundamental frequency, whereas the other L−1 oscillations areits harmonics. The value of L is fixed upon exploratory data analysis,such as through empirical testing/experimentation. The coefficientsa_(1,t)(s_(i)), . . . , a_(2L+1,t)(s_(i)) depend on the spatial locationof the observation and also change over time. The factor b_(f) isemployed to dampen the temporal variability of high-frequencyoscillations (through γ₀ and γ₁), (Equation 3.0):b _(f)=(γ₀+(1−γ₀)×exp[−γ₁(f−1)]),Where γ₀∈[0,1], γ₁∈

⁺, and v∈

are unknown parameters.

To make the model parsimonious, able to cope with large N and capable ofperforming spatial interpolation, the coefficients are mapped onto a setof gridded, latent Gaussian Processes (Equation 4.0):

${{a_{jt}( s_{i} )} = {\sum\limits_{g = 1}^{G}{{k\lbrack {s_{i},s_{g}^{*}} \rbrack} \times {\theta_{jt}( s_{g}^{*} )}}}},{i \in \{ {1,\ldots\mspace{11mu},N} \}},{j \in \{ {1,\ldots\mspace{11mu},P} \}},{g \in \{ {1,\ldots\mspace{11mu},G} \}},$Where k[s_(i), s*_(g)]∈[0,1] is a kernel evaluation, described furtherin Section 4.3, and θ_(jt)(s*_(g)), t∈{1, 2, . . . , T} is a latentGaussian process, for any grid point s*_(g) and component j. Theconstant G denotes the number of points on a grid that is superimposedon the spatial domain under study.

Hence, in every grid point, there exists P=2L+1 latent Gaussianprocesses.

In some embodiments, the temporal evolution of the Gaussian Processes ismodeled as a random walk:

${{\theta_{jt}( s_{g}^{*} )} = {{\theta_{j,{t - 1}}( s_{g}^{*} )} + {ɛ_{jt}( s_{g}^{*} )}}},{{ɛ_{jt}( s_{g}^{*} )}\mspace{11mu}\text{∼}{{{Normal}\mspace{11mu}\lbrack {0,\frac{c_{t}( s_{g}^{*} )}{\delta}} \rbrack}.}}$In the expression above, the unknown parameter δ∈(0,1], also known as adiscount factor, controls the decay of information over time. Furtherdetails regarding the evolution variance c_(t)(s*_(g)), for t∈(0, . . ., T) and j∈{1, . . . , P} is provided in Section 4.7 (“Covariance”). Theevolution equation above allows for time and space varying amplitudes inthe seasonal components and requires initial conditions, namely(Equation 5.0):θ_(j,0)(s* _(g))˜Normal [m _(j,0) ,c ₀(s* _(g))].

4.3 Spatial Variability

In this section a means to generate the kernel evaluations k[s_(i),s*_(g)] which connect the coefficient a_(j,t)(s_(i)) to the G latent,gridded Gaussian processes θ_(jt)(s_(i)), . . . , θ_(jt)(s*_(G)) fori∈{1, . . . , N} and j∈{1, . . . , P} (See Equation 4.0). In anembodiment, Discrete Process Convolutions is used to generate the kernelevaluations.

As mentioned previously in Section 4.2 (“Temporal Variability”), supposethe spatial domain of interest is overlaid with a grid that has a totalof G Points. To each grid point, a vector of P latent Gaussian Processesis assigned. The Mahalanobis distance between the observation i (i∈{1, .. . , N}) and grid point g (g∈{1, . . . , G}), respectively located ats_(i) and s*_(g), is provided by∥s _(i) −s* _(g)∥Σ₁=√{square root over (d′ _(ig)Σ_(i) ⁻¹ d _(ig))}

Where d_(ig)=(Δx(s_(i), s*_(g)), Δy(s_(i), s*_(g))) is the vector oflongitudinal and latitudinal distances between s_(i) and s*_(g). In someembodiments, the 2×2 positive definite matrix Σ_(i) is alocation-independent, fixed multiple of the identity matrix Σ_(i)=4r²I₂,where r is the grid resolution. Thus, the technique works with Euclideandistances, where the contribution of each observation is weighted basedon the distance from that observation to each of the gridded pointsalong the grid superimposed on the spatial domain. The unnormalizedkernel associated with this pair of locations is (Equation 6.0):

  k^(un)[s_(i), s_(g)^(*)] = (1 − min [1, s_(i) − s_(g)^(*)])².

Under the choice

${{\sum\limits_{i}^{\;}{= {4r^{2}I_{2}}}},{{d_{ig}}_{\sum\limits_{i}^{\;}\;} = {1{\mspace{11mu}\;}{and}}}}\mspace{14mu}$k^(un)[s_(i), s_(g)^(*)] = 0is obtained whenever ∥d_(ig)∥=2r. In other words, the kernel's rangecorresponds to twice the grid resolution. In the G×1 vector ofunnormalized kernel evaluations associated with location s_(i),k^(un)(s_(i))=(k^(un)[s_(i), s*₁], . . . , k^(un) [s_(i), s*_(G)])′, thenumber of non-zero elements does not exceed 14, regardless of thelocation s_(i) relative to the latent grid.

The normalized kernel is obtained by considering all G evaluations ofthe kernel centered at s_(i) (Equation 7.0):

${k\lbrack {s_{i},s_{g}^{*}} \rbrack} = {\frac{k^{un}\lbrack {s_{i},s_{g}^{*}} \rbrack}{\sum\limits_{a = 1}^{G}{k^{un}\lbrack {s_{i},s_{a}^{*}} \rbrack}}.}$Hence, guaranteeing that k[s_(i), s*_(g)]∈[0,1] and Σ_(g=1) ^(G)k[s_(i),s*_(g)]=1, for all i. Owing to this, the coefficient a_(j,t)(s_(i)) is aconvex combination of the latent processes θ_(jt)(s_(i)), . . . ,θ_(jt)(s*_(G)).

4.4 Multivariate Model

Equation 2.0 may be written more succinctly. If d(t) is written as theday of year to which instant t belongs (for exampled(1)=d(366)=d(−364)=1), and define 365 vectors of size P×1 (Equation8.0):w _(h)=(1,s _(1,h) ,c _(1,h) , . . . ,s _(L,h) ,c _(L,h))′,h∈{1,2, . . .,365},where

$\mspace{20mu}{{s_{f,t} = {b_{f} \times {\sin\lbrack \frac{2f\;\pi\; t}{365} \rbrack}}},\mspace{20mu}{c_{f,t} = {b_{f} \times {\cos\lbrack \frac{2f\;\pi\; t}{365} \rbrack}}},}$

for f∈{1, . . . , L}. Also, let k(s_(i))=(k[s_(i), s*₁], . . . ,k[s_(i), s*_(G)]), denote the G×1 vector of normalized kernelevaluations associated with location s_(i), let (θ_(jt)(s*₁), . . . ,θ_(jt)(s*_(G)))′ represent the G×1 random vector associated withcomponent j, and let these P vectors be stacked into θ_(t)∈

^(GP×1). Then, using a Kronecker product, Equation 2.0 may be rewrittenas (Equation 9.0):x _(t)(s _(i))=(w _(d(t)) ⊗k(s _(i)))′θ_(t).Moreover, if there exists a vector of N observations, all collected onday t, it may be written as,T _(t) ^(raw)=(Y _(t) ^(raw)(s ₁), . . . ,Y _(t) ^(raw)(s _(N)))′,and expand Equation 1.0 to a multivariate model (Equation 10.0):

${\Upsilon_{t}^{raw} = {x_{t} + \zeta_{t}}},{\zeta_{t}\text{\textasciitilde}{{{Normal}\lbrack {0,{\frac{\upsilon^{2}}{\tau}I_{N}}} \rbrack}.}}$To explicitly write the random vector θ_(t), the model may bereformulated asY _(t) =F _(t)θ_(t)+∈_(t),whereY _(t)=√{square root over (τ)}×(Y _(t) ^(raw)(s ₁), . . . ,Y _(t)^(raw)(s _(N)))′,the residuals are i.i.d. (independent and identically distributed)Gaussian,∈_(t)˜Normal [0,v ² I _(N)]and matrix F_(t)∈

^(N×GP) can be constructed with a N×G matrix of scaled kernelevaluationsK=√{square root over (τ)}×(k(s ₁), . . . ,k(s _(N)))′,Together with the P×1 vector w′_(g)(t) (Equation 11.0):F _(t) =w′ _(d(t)) ⊗K

4.5 Point Measurements Vs. Areal Averages

In the previous sections, a general framework was provided for amultivariate model. Consider a case where there are two types of data:point measurements and areal averages. In an embodiment, to capture theareal averages, integrals are approximated with local quadraticpolynomials, where the coefficients are based on the fit provided byDiscrete Process Convolutions described above.

Without loss of generality, consider N=2, so that the 2×1 vector of rawobservations Y_(t) ^(raw)(S), and a single point measurement, Y_(t)^(raw)(s). In other words, s is a point in the domain of interest,whereas S is an area in the same domain; therefore Y_(t) ^(raw)(⋅)corresponds to a point measurement or an areal average, depending on theargument in the parenthesis. Similarly to Equation 1.0 and Equation10.0, Y_(t) ^(raw)(S) and Y_(t) ^(raw)(s) are modeled as (Equation12.0):

${\begin{pmatrix}{\Upsilon_{t}^{raw}(S)} \\{\Upsilon_{t}^{raw}(s)}\end{pmatrix} = {\begin{pmatrix}{x_{t}(S)} \\{x_{t}(s)}\end{pmatrix} + \zeta_{t}}},$where

${\zeta_{t}\text{\textasciitilde}{{Normal}\lbrack {0,\begin{pmatrix}\upsilon^{2} & 0 \\0 & \frac{\upsilon^{2}}{x}\end{pmatrix}} \rbrack}},$(x_(t)(S), x_(t)(s)) is a vector of “noise-free measurements”, v² is thevariance associated with the areal average data set, and τ is a factorthat, together with v², controls point measurement precision (forexample, with τ=1, the two data sources have the same precision).

The spatial and temporal variation x_(t)(s) is provided by Equation 9.0.On the other hand, for areal average x_(t)(S), the followingapproximation is adopted:x _(t)(S)=∫_(S) x _(t)(s)ds≈n′x _(t)

,where the 5×1 vector n is constructed as

${n = ( {\frac{8}{12},\frac{1}{12},\frac{1}{12},\frac{1}{12},\frac{1}{12}} )^{\prime}},$and the 5×1 vector x_(t)

contains the values of x_(t) at the center (“ce”), of the area S, aswell as the upper left (“ul”), upper right (“ur”), lower left (“ll”) andlower right (“lr”) corners of S:x _(t)

=(x _(t)(s ^(ce))x _(t)(s ^(ul)),x _(t)(s ^(ur)),x _(t)(s ^(ll)),x_(t)(s ^(lr)))′.Hence,x _(t)(S)≈(w _(t) ⊗k(S))′θ_(t),Where the G×1 vector k(S) is provided by (Equation 13.0)

$\begin{matrix}{{k(S)} = {n^{\prime}\begin{pmatrix}{k( s^{ce} )} \\{k( s^{ul} )} \\{k( s^{ur} )} \\{k( s^{ll} )} \\{k( s^{lr} )}\end{pmatrix}}} \\{= {{\frac{8}{12}{k( s^{ce} )}} + {\frac{1}{12}( {{k( s^{ul} )} + {k( s^{ur} )} + {k( s^{ll} )} + {k( s^{lr} )}} )}}}\end{matrix}.$In other words, the g-th element of this vector consists of the sum

${k_{g}(S)} = {{\frac{8}{12}{k( {s^{ce},s_{g}^{*}} )}} + {\frac{1}{12}{\{ {{k( {s^{ul},s_{g}^{*}} )} + {k( {s^{ur},s_{g}^{*}} )} + {k( {s^{ll},s_{g}^{*}} )} + {k( {s^{lr},s_{g}^{*}} )}} \}.}}}$

In embodiments where isotropic kernels with Σ_(i)=4r²I₂ are employedtogether with a discrete process convolution grid of the same resolutionas the gridded data set. As a result, Equation 13.0 simplifiesconsiderably. If the convolution grid is coarser than the gridded dataset, no problems would emerge and no additional steps would be needed.However, if the convolution grid is finer than the gridded data set,then it would be more difficult to compute the model's mean temperatureacross a data grid cell. A fine convolution grid may be used incircumstances where a large number of station observations wereavailable. Coarse convolution grids will lose the fine details of rapidtemperature variation across space, which may be undesirable in somecases.

4.6 Model Summary

To summarize the hierarchical model constructed up to this point, themultivariate observation layer of the model is (Equation 14.0):(Y _(t)|θ_(t),τ)˜Normal [F _(t)θ_(t) ,v ² I _(N)],where Y_(t)∈

^(N×1) is a vector of scaled observations, which contains N_(a) arealaverages and N_(p)=N−N_(a) scaled point measurements:Y _(t) ={Y _(t) ^(raw)(S ₁), . . . ,Y _(t) ^(raw)(s _(N) _(g) ),√{squareroot over (τ)}×Y _(t) ^(raw)(S ₁), . . . ,√{square root over (τ)}×Y _(t)^(raw)(s _(N) _(p) )}′.Matrix F_(t)∈

^(N×GP) is built according to Equation 11.0, it includes temporalcomponents and spatially varying “weights”, respectively through vectorw_(d(t))∈

^(P×1) and matrix K∈

^(N×G). Equation 8.0 describes w_(d(t)), whereas K is formed asK={k(S ₁), . . . ,(S _(N) _(g) ),√{square root over (τ)}×k(s ₁), . . .,√{square root over (τ)}×k(s _(N) _(p) ))}′.In this expression, the kernel vectors for areal averages are providedby Equation 13.0. Equations 6.0 and 7.0 describe how kernel evaluationsare computed and normalized. As an example, the rows of K thatcorrespond to area S₁ and location s₁, sized 1×G, respectivelycorrespond to

$\begin{pmatrix}{{\frac{8}{12}{k\lbrack {S_{1}^{ce},s_{1}^{*}} \rbrack}} + {\frac{1}{12}( {{k\lbrack {S_{1}^{ul},s_{1}^{*}} \rbrack} + {k\lbrack {S_{1}^{ur},s_{1}^{*}} \rbrack} + {k\lbrack {S_{1}^{ll},s_{1}^{*}} \rbrack} + {k\lbrack {S_{1}^{lr},s_{1}^{*}} \rbrack}} )}} \\{{\frac{8}{12}{k\lbrack {S_{1}^{ce},s_{2}^{*}} \rbrack}} + {\frac{1}{12}( {{k\lbrack {S_{1}^{ul},s_{2}^{*}} \rbrack} + {k\lbrack {S_{1}^{ur},s_{2}^{*}} \rbrack} + {k\lbrack {S_{1}^{ll},s_{2}^{*}} \rbrack} + {k\lbrack {S_{1}^{lr},s_{2}^{*}} \rbrack}} )}} \\\vdots \\{{\frac{8}{12}{k\lbrack {S_{1}^{ce},s_{G}^{*}} \rbrack}} + {\frac{1}{12}( {{k\lbrack {S_{1}^{ul},s_{G}^{*}} \rbrack} + {k\lbrack {S_{1}^{ur},s_{G}^{*}} \rbrack} + {k\lbrack {S_{1}^{ll},s_{G}^{*}} \rbrack} + {k\lbrack {S_{1}^{lr},s_{G}^{*}} \rbrack}} )}}\end{pmatrix}^{\prime}$ and$\sqrt{\tau} \times {( {{k\lbrack {s_{1},s_{1}^{*}} \rbrack},{k\lbrack {s_{1},s_{2}^{*}} \rbrack},\ldots\mspace{14mu},{k\lbrack {s_{1},s_{G}^{*}} \rbrack}} ).}$

Parameter τ defines the precision of station measurements, relative tothat of areal estimates. The random vector θ_(t)∈

^(GP×1) controls the baseline levels plus the amplitudes and phases ofthe L intra-annual oscillations. These oscillations are not stationary:rather, θ_(t) can evolve as a multivariate random walk, with discountfactor δ (Equation 15.0):

$( {{\theta_{t}❘\theta_{t - 1}},\delta} )\text{\textasciitilde}{{Normal}\lbrack {\theta_{t - 1},{\frac{1}{\delta}P_{t - 1}^{- 1}}} \rbrack}$The large precision matrix P_(t)∈

^(GP×GP) discussed later in Section 4.7. Finally the 2×1 vector (γ₀,γ₁), which enters the computation of w_(d(t)) (see Equation 3.0 andEquation 8.0), defines the decay of evolution variance, from low to highfrequencies.

The model is completed by defining the initial distribution of thelatent state (Equation 16.0),θ₀˜Normal [m ₀ ,P ₀ ⁻¹],Which requires specifying the initial mean m₀∈

^(GP×GP) and the initial precision P_(0∈)

^(GP×GP). This task is explained below in Section 4.8. Note that thediagonal elements in the covariance matrix P₀ ⁻¹ are referred to asc₀(s*₁), . . . , c₀(s*_(G)), in Equation 5.0.

Conceptually, the resolution of the discrete process convolution gridcan be seen as a free parameter; the same is true for the horizontal andvertical offsets relative to the coordinate axes. In practice, however,it is computationally unfeasible to treat these parameters as continuousand attempt to estimate them like the others. A potential workaround isto choose a few configurations and perform model selection.

4.7 Covariance

According to a theorem proposed by West and Harrison (BayesianForecasting and Dynamic Models. Springer Series in Statistics. SpringerNew York, 1999), for any observable constant Dynamic Linear Model(DMLs), the limiting variance lim_(t→∞)C_(t)=C exists and is independentof the initial information. Here, this theorem is applied to explore thelimiting (or asymptotic) covariance structure of the model, which fallsin the category of observable periodic DLMs. Therefore, instead ofhaving a single limiting covariance, the model has 365, as this numbercorresponds to the longest period of the oscillations in the observationmatrix F_(t).

According to the DLM forward filtering method (again proposed by Westand Harrison), the posterior precision at time t is provided by

$P_{t} = {{\delta\; P_{t - 1}} + {\frac{1}{\upsilon^{2}}F_{d{(t)}}^{\prime}{F_{d{(t)}}.}}}$The recursion can be expanded as:

$\begin{matrix}{P_{t} = {{\delta^{2}P_{t - 2}} + {\frac{\delta}{\upsilon^{2}}F_{d{({t - 1})}}^{\prime}F_{d{({t - 1})}}} + {\frac{1}{\upsilon^{2}}F_{d{(t)}}^{\prime}F_{d{(t)}}}}} \\{= \ldots} \\{= {{\delta^{365}P_{t - 365}} + {\frac{1}{\upsilon^{2}}{\sum\limits_{a = 0}^{364}{\delta^{a}F_{d{({t - a})}}^{\prime}F_{d{({t - a})}}}}}}} \\{= {{\delta^{365}P_{t - 365}} + {\frac{1}{\upsilon^{2}}{\sum\limits_{a = 0}^{364}{{\delta^{a}( {w_{d{({t - a})}}w_{d{({t - a})}}^{\prime}} )} \otimes ( {K^{\prime}K} )}}}}}\end{matrix}$

If it is assumed that t is large and makes use of the theorem, thenP_(t)=P_(t−365)=P_(d(t)), regardless of the day of the year to whichthat t corresponds. As a result, the expression for this limitingprecision is (Equation 17.0):

$P_{d{(t)}} = {\frac{1}{\upsilon^{2}}{X_{d{(t)}} \otimes ( {K^{\prime}K} )}}$where X_(d(t))∈

^(P×P) is provided by

$X_{d{(t)}} = {\frac{1}{1 - \delta^{365}}{\sum\limits_{a = 0}^{364}\;{\delta^{a}w_{d{({t - a})}}{w_{d{({t - a})}}^{\prime}.}}}}$Unless δ∈(0.98, 1), it is safe to approximate δ³⁶⁵≈0. The asymptoticposterior covariance is given by (Equation 18.0):

$\begin{matrix}{C_{d{(t)}} = P_{d{(t)}}^{- 1}} \\{= {\upsilon^{2}{X_{d{(t)}}^{- 1} \otimes ( {K^{\prime}K} )^{- 1}}}}\end{matrix}.$

4.8 Initial Posterior Mean States

Let

₀=Ø denote the initial information and

_(t)={

_(t−1), Y_(t)}, for t=1, . . . , T. If the asymptotic matrix P_(d(0)) isused to specify the initial distribution, (θ₀|

₀)˜Normal [m₀, P₀ ⁻¹], a model is obtained where all the covariancematrices have converged to their asymptotic values for the very firstinstant. This strategy also has implications in the computation ofposterior means, which is explored in this section.

According to the DLM forward filtering algorithm, the equation for theposterior mean at time t, m_(t)∈

^(GP×1), is (Equation 19.0):

$\begin{matrix}{m_{t} = {P_{t}^{- 1}( {{\delta\; P_{t - 1}m_{t - 1}} + {\frac{1}{\upsilon^{2}}F_{t}^{\prime}Y_{t}}} )}} \\{= {{\delta\; P_{t}^{- 1}P_{t - 1}m_{t - 1}} + {\frac{1}{\upsilon^{2}}P_{t}^{- 1}F_{t}^{\prime}Y_{t}}}}\end{matrix}.$Since limiting covariances are used throughout, Equation 17.0 mayreplace P_(t−1) and P_(t), regardless of t, and ease the computationalcomplexity:m _(t)=δ(X _(d(t)) ⁻¹ X _(d(t−1)) ⊗I _(G))m _(t−1) +X _(d(t)) ⁻¹ w_(d(t))⊗((K′K)⁻¹ K′Y _(t).The initial mean, m₀, may be equated to the solution of the linearsystem that employs the first year of data as follows:

$\begin{pmatrix}\Upsilon_{1} \\\Upsilon_{2} \\\vdots \\\Upsilon_{365}\end{pmatrix} = {{\begin{pmatrix}F_{1} \\F_{2} \\\vdots \\F_{365}\end{pmatrix}m_{0}} + \begin{pmatrix}\epsilon_{1} \\\epsilon_{2} \\\vdots \\\epsilon_{365}\end{pmatrix}}$Where errors are i.i.d. Normal. In other words, this is comparable to asimplified version of Equation 2.0, where the coefficients a₁ (s), . . ., a_(2L+1)(s) are time-invariant:

${x_{t}( s_{i} )} = {{a_{1}( s_{i} )} + {\sum\limits_{f = 1}^{L}\;{b_{f} \times {\{ {{{a_{2\; f}( s_{i} )}{\sin\lbrack \frac{2\pi\;{ft}}{365} \rbrack}} + {{a_{{2\; f} + 1}( s_{i} )}{\cos\lbrack \frac{2\pi\;{ft}}{365} \rbrack}}} \}.}}}}$These coefficients are Gaussian variables, located on the same grid asthe original Gaussian processes. Their prior distribution is flat,meaning that their posterior mean equates to the least squares solutionof the system described above.

4.9 Log-Likelihood Function

Ψ denotes the collection of all time-invariant parameters in the mode:the discount factor δ, the decay parameters γ₀ and γ₁, and theseasonality parameter v. Full Bayesian inference on Ψ can be performedby specifying appropriate priors and exploring the posterior via MCMCmethods. An easier approach comprises specifying uniform priors, so thatboundary constraints are satisfied (for example γ₀∈[0,1]), andmaximizing the posterior density, which is tantamount to maximizing thelikelihood. In state-space models, the likelihood can be computed as theproduct of the one-step forecast densities, via successive applicationof the Bayes rule:p(Y ₁ ,Y ₂ , . . . Y _(T)|ψ)=p(Y _(T) |D _(T−1),ψ)×p(Y _(T−1) |D_(T−2),ψ)× . . . ×p(Y ₁ |D ₀,ψ).Thus, the log-likelihood of the entire data set is

${\ell_{\Psi}( {Y_{1},Y_{2},\ldots\mspace{14mu},Y_{T}} )} = {{\sum\limits_{t = 1}^{T}\;\ell_{\Psi,t}} = {\sum\limits_{t = 1}^{T}\;{\log\;{{p( { Y_{t} \middle| D_{t - 1} ,\Psi} )}.}}}}$To keep the notation simple, the dependence of matrices and vectors on Ψis omitted. The one-step forecast distribution at time t is(Y _(t) |D _(t−i),Ψ)˜Normal [h _(t) ,B],meaning that (Equation 20.0):

${p( { Y_{t} \middle| D_{t - 1} ,\Psi} )} = {\frac{1}{( {2\pi} )^{\frac{N}{2}}{B}^{\frac{1}{2}}\tau^{\frac{N_{p}}{2}}} \times {{\exp\lbrack {{- \frac{1}{2}}( {Y_{t} - h_{t}} )^{\prime}{B^{- 1}( {Y_{t} - h_{t}} )}} \rbrack}.}}$In these expressions (Equation 21.0):h _(t) =F _(t) m _(t−1),The factor τ^(N) ^(P) ^(/2) is the Jacobian of the transformationoperated on the raw data, and the large covariance matrix B∈

^(N×N) is provided by (Equation 22.0):

$B = {{\frac{1}{\delta}F_{t}P_{t - 1}^{- 1}F_{t}^{\prime}} + {\upsilon^{2}{I_{N}.}}}$

At first, it would seem that matrix B depends on t. However, this is notthe case because of the use of the limiting precision matrices. First,employ the QR decomposition of matrix K∈

^(N×G):

$K = {{( {Q\mspace{14mu} Q^{\bot}} )\begin{pmatrix}R \\0\end{pmatrix}} = {{QR}.}}$In this expression, Q∈

^(N×G) is the orthonormal rectangular matrix, Q⁻¹∈

^(N×(N−G)) is the orthogonal complement of Q,R∈

^(G×G), upper triangular square matrix, and 0 is a (N−G)×G matrix ofzeroes. Using the definitions of F_(t) (Equation 11.0) and P_(d(t))(Equation 17.0), Equation 22.0 is modified into:

$\begin{matrix}{B = {{\frac{1}{\delta}( {w_{d{(t)}}^{\prime} \otimes K} )( {\frac{1}{\upsilon^{2}}{X_{d{({t - 1})}} \otimes ( {K^{\prime}K} )}} )^{- 1}( {w_{d{(t)}}^{\prime} \otimes K} )^{\prime}} + {\upsilon^{2}I_{N}}}} \\{= {\upsilon^{2}( {{\frac{1}{\delta}w_{d{(t)}}^{\prime}X_{d{({t - 1})}}^{- 1}{w_{d{(t)}} \otimes {K( {K^{\prime}K} )}^{- 1}}K^{\prime}} + I_{N}} )}} \\{= {\upsilon^{2}( {{{\frac{1 - \delta^{P}}{\delta^{P}} \otimes {{QR}( {R^{\prime}Q^{\prime}{QR}} )}^{- 1}}R^{\prime}Q^{\prime}} + I_{N}} )}} \\{= {{\upsilon^{2}( {{\frac{1 - \delta^{P}}{\delta^{P}}{QQ}^{\prime}} + I_{N}} )}.}}\end{matrix}$

In order to compute the determinant and the inverse of B, it is usefulto obtain its spectral decomposition. First, because of thecomplementary Q and Q^(⊥),

${( {Q\mspace{14mu} Q^{\bot}} )\begin{pmatrix}Q^{\prime} \\( Q^{\bot} )^{\prime}\end{pmatrix}} = {{{Q\; Q^{\prime}} + {Q^{\bot}( Q^{\bot} )}^{\prime}} = {I_{N}.}}$Then, to obtain the spectral decomposition:

$B = {{\upsilon^{2}( {Q\mspace{14mu} Q^{\bot}} )}\begin{pmatrix}{\frac{1}{\delta^{P}}I_{G}} & 0 \\0 & I_{N - G}\end{pmatrix}\begin{pmatrix}Q^{\prime} \\( Q^{\bot} )^{\prime}\end{pmatrix}}$Given this, det[B]=|B|=v^(2N)δ^(−GP) and

$\begin{matrix}{B^{- 1} = {\frac{1}{\upsilon^{2}}( {Q\mspace{14mu} Q^{\bot}} )\begin{pmatrix}{\delta^{P}I_{G}} & 0 \\0 & I_{N - G}\end{pmatrix}\begin{pmatrix}Q^{\prime} \\( Q^{\bot} )^{\prime}\end{pmatrix}}} \\{= {\frac{1}{\upsilon^{2}}{( {I_{N} - {( {1 - \delta^{P}} ){QQ}^{\prime}}} ).}}}\end{matrix}$By plugging the determinant of B into the one-step forecast density attime t (using Equation 20.0),

${{p( { Y_{t} \middle| D_{t + 1} ,\Psi} )} = {( {2\pi} )^{{- N}/2}\upsilon^{- N}{\delta^{{{GP}/2}\tau}}^{{- N_{p}}/2}{\exp\lbrack {{- \frac{1}{2}}e_{t}^{\prime}B^{- 1}e_{t}} \rbrack}}},$Where e_(t) denotes the N×1 vector of residuals at time t:e_(t)=Y_(t)−h_(t).

Let z_(t)=Q′Y_(t) and a_(t)=Q′e_(t) denote the G×1 vectors oftransformed observations and residuals, respectively. Then, we may writethe log-likelihood l_(Ψ,t) economically as

$\ell_{\Psi,f} = {{- \frac{1}{2}}{( {{N\;{\log( {2{\pi\upsilon}^{2}} )}} - {{GP}\;{\log(\delta)}} + {N_{p}{\log(\tau)}} + {\frac{1}{\upsilon^{2}}{\sum\limits_{i = 1}^{N}\; Y_{i\; f}^{2}}} + {\frac{1}{\upsilon^{2}}{\sum\limits_{g = 1}^{G}\;( {{\delta^{P}a_{gt}^{2}} - z_{gt}^{2}} )}}} ).}}$This expression can be used to sample from the posterior distribution ofthe model parameters in a fully Bayesian implementation.

4.10 Missing Observations

Weather station data sets commonly have missing values. This issue canbe addressed by filling in those values with the corresponding one stepforecast means, using Equation 21.0. For example, if Y_(t) ^(raw)(s_(i))is missing, it can be replaced with h_(t)(s_(i)). Under this strategy,the corresponding residual is zero (θ_(t)(s_(i))=0) and the number ofobservations is constant regardless of t.

4.11 Hindcasts and Reforecasts

One of the most important functions of a statistical weather model is togenerate probabilistic descriptions of the weather, at locations orinstants where information was not provided to the mode. In a Bayesiancontext, these are called posterior predictive distributions and areextremely valuable, since they form the basis of many validationtechniques. In this section and the following, the process through whichthose distributions can be derived from the proposed model after themodel has been fitted to the data is described.

Assume a set of N_(f) locations s*₁, . . . s*_(N) _(f) , which need notcoincide with the original set s₁, . . . , s_(N). Given all theinformation available up to time t, the goal is to obtain thedistribution of Y*_(t) (i.e., hindcasts) and Y*_(t+1), Y*_(t+2), . . .(i.e., re-forecasts).

Under the model specified earlier, the distribution of these randomvectors is(Y* _(t+1) |D _(t))˜Normal [h* _(t+l) ,B* _(t+1)],l=0,1, . . .such that

${h_{i + 1}^{*} = {( {w_{d{({t + 1})}}^{\prime} \otimes K^{*}} )m_{t}}},\begin{matrix}{B_{t + 1}^{*} = {{\frac{1}{\delta^{1}}( {w_{d{({t + 1})}}^{\prime} \otimes K^{*}} ){C_{d{(t)}}( {w_{d{({t + 1})}}^{\prime} \otimes K^{*}} )}^{\prime}} + {\upsilon^{2}I_{N_{f}}}}} \\{= {{x_{t,l}{K^{*}( {K^{\prime}K} )}^{- 1}( K^{*} )^{\prime}} + {\upsilon^{2}I_{N_{f}}}}} \\{= {{{x_{t\; l}( {K^{*}R^{- 1}} )}( {K^{*}R^{- 1}} )^{\prime}} + {\upsilon^{2}I_{N_{f}}}}}\end{matrix}$where K*∈

^(N) ^(f) ^(×G) is the kernel matrix for the locations where thehindcast/re-forecast is to be computed,

$x_{t,l} = {\frac{\upsilon^{2}}{\delta^{1}}w_{d{({t + 1})}}^{\prime}X_{d{(t)}}^{- 1}{w_{d{({t + l})}}.}}$and C_(d(t)) is defined in Equation 18.0.

Due to the discount factors (0<δ≤1), the variance B*_(t+1) growsexponentially with lead l. This can be undesirable for long-termforecasting, but may be mitigated by setting δ≈1. The penalty associatedwith this measure is that the model may become overly smooth in time.

4.12 Backward Smoothing

Another potential use is to discover the historical states of theGaussian processes given observations up to the present. These can bedescribed by the smooth distribution (θ|D_(T))˜Normal({tilde over(m)}_(t), {tilde over (C)}_(t)), where {tilde over (m)}_(t)∈

^(GP×1) and C_(t)∈

^(GP×GP) can be calculated recursively from{tilde over (m)} _(t)=(1−δ)m _(t) +δ{tilde over (m)} _(t+1){tilde over (C)} _(t)=(1−δ)C _(t)+δ² {tilde over (C)} _(t+1)fort−T−1,T−2, . . . ,1, and {tilde over (m)} _(T) ={tilde over (m)} _(T),{tilde over (C)} _(T) =C _(T)

The retrospective descriptions of the states of processes enablesestimation at locations of interest that are not described by the dataset, at any historical time. These locations can be anywhere in thespatial domain, provided that at least one discrete process convolutiongrid point is within their range. Locations may be on a grid or not,global or regional, and the grid resolution may be higher than, equalto, or lower than the original resolution of the climate model output.

The posterior predictive distribution for quantities at N_(s) locationsof interest, {tilde over (Y)}_(t) is({tilde over (Y)} _(t) |D _(T))˜Normal [{tilde over (h)} _(t) ,B _(t)],where {tilde over (h)}_(t)∈

^(N) ^(s) ^(×1) and B_(t)∈

^(N) ^(s) ^(×N) ^(s) are given by{tilde over (h)} _(t)=(w′ _(t) ⊗K){tilde over (m)} _(t),{tilde over (B)} _(t)=(w′ _(t) ⊗{tilde over (K)}){tilde over (C)}_(t)({tilde over (K)}⊗w _(t)),and {tilde over (k)}∈

^(N) ^(s) ^(×G) is a kernel matrix for locations where the posteriorpredicative distribution is to be computed.

4.13 Backward Sampling

In some cases, it is beneficial to obtain simulations of the latentprocesses, such as for sample-based studies. Under the dynamic linearmodeling framework, it is possible to sample the vectors θ₁, . . . ,θ_(T) from their joint posterior distribution, p(θ₁, . . . ,θ_(T)|D_(T)), using a backward sampling algorithm.

At time t, the backward sampling variance is:C _(t) =u _(t) p _(t) ⁻¹ =u _(t) v ² X _(m(t)) ⁻¹⊗(K′K)⁻¹where

$u_{t} = \{ {\begin{matrix}{1,} & {{{if}\mspace{14mu} t} = T} \\{( {1 - \delta} ),} & {{{if}\mspace{14mu} t} \in \{ {1,\ldots\mspace{14mu},{T - 1}} \}}\end{matrix},} $and the backward sampling mean is

${\overset{.}{m}}_{i} = \{ {\begin{matrix}{m_{T},} & {{{if}\mspace{14mu} t} = T} \\{{{( {1 - \delta} )m_{t}} + {\delta{\overset{.}{\theta}}_{t + 1}}},} & {{{if}\mspace{14mu} t} \in \{ {1,\ldots\mspace{14mu},{T - 1}} \}}\end{matrix}.} $Given the above, the backward sampling distribution for θ_(1:T−1) iswritten as(θ_(t) |D _(t),θ_(t+1))˜Normal [m _(t) ,u _(t) v ² x _(m(t))⁻¹⊗(K′K)⁻¹].

A fast-backward sampling algorithm is as follows:

1. For d ∈ {1,2,...,365},  (a) compute U_(d) as the upper Choleskyfactor of X_(d), i.e., X_(d) =   U′_(d)U_(d);  (b) compute O_(d(t)) =U_(d(t)) ⊗ R 2. For t ∈ {T,T−1,...,1} do  (a) Sample {grave over(z)}_(t) ~ Normal [0,I_(GP)];  (b) Set = {grave over (θ)}₁ = {squareroot over (u_(t)v²)}O_(d(t)) ⁻¹{grave over (z)}_(t) + {grave over(m)}_(t) 3. Return {grave over (Θ)}_(T) = {{grave over (θ)}₁,{grave over(θ)}₂,...,{grave over (θ)}_(T)}The ability to generate ensembles of Θ_(T), which include spatial andtemporal dependencies, helps determine the uncertainty quantification.

5.0 Data Blending Process Flow

FIG. 6 illustrates a process for fitting a model that blends point datawith areal averages in block diagram form according to an embodiment.FIG. 6 represents an example of the aforementioned process and may varybetween different embodiments. For example, the steps recited in theblocks of FIG. 6 may be divided out into multiple sub-steps, combinedinto a smaller set of steps, performed in a different order, and soforth in other embodiments. In order to illustrate clear examples, thefollowing description assumes that the process flow of FIG. 6 isperformed by the data blending subsystem 170.

In FIG. 6, at block 600 the weather data input module 171 receivesobservation data 501 for an environmental variable, which includes pointdata and areal averages, at corresponding times and locations. Examplesof observations that may be obtained and sources from which the weatherdata input module 171 obtains the observation data 501 is described inmore detail above in Section 3.0 “Example System Inputs”. In someembodiments, the weather data input module 171 is configured to receiveuser input from the field manager computing device 104 that specifiesthe source(s) from which to obtain the observation data 501 and/or thetype of environmental variable the observation data 501 represents. Inother embodiments, the data blending subsystem 170 is invoked by anotherapplication (not depicted in FIG. 1) that supplies the sources fromwhich to obtain the observation data 501. In yet other embodiments, theweather data input module 171 reads a configuration file that specifieswhere to obtain the observation data 501.

At block 601, the model fitting module 172 defines a state-space modelthat represents the behavior of the environmental variable across spaceand time. In some embodiments, the state-space model includes anobservation equation that describes how observations from differentsources relate to gridded, latent Gaussian processes, a state equationthat describes how the Gaussian processes evolve over time, and aninitial state equation that describes the mean and variance of theGaussian processes at an initial state used to drive the model.

In an embodiment, the observation equation appears asY_(t)=F_(t)θ_(t)+∈_(t), where Y_(t) is a vector of observationscollected at time t∈{1, . . . , T} at multiple locations; F_(t) is amatrix that maps those observations to a vector of latent Gaussianprocesses θ_(t); and ∈_(t) is a vector of Gaussian measurement errors,∈_(t)˜N [0, v²I]. The observation equation describes how observationsfrom different sources (for example point measurements, areal averages)relate to gridded, latent Gaussian processes. Matrix F_(t) isresponsible for this mapping. In some embodiments, each entry in F_(t)is constructed by evaluating a function (kernel) that only takesnon-zero values within a predefined spatial range. As a result F_(t) isa sparse matrix, meaning that computations involving F_(t) can beperformed quickly using techniques such as sparse matrix multiplication.The kernel function used depends on whether the correspondingobservation is a point measurement or an areal average. The non-zeroentries of matrix F_(t) are multiplied by several temporal harmonics(i.e., sines and cosines with varying periods), so that the model isable to capture seasonal variability. Thus, the latent Gaussianprocesses θ capture the spatial and temporal variation of the seasonalcycles. Finally, measurement error is captured through vector ∈_(t). Insome embodiments, for simplicity, the aforementioned errors are assumedto be spatially and temporally uncorrelated. In some embodiments, thevariance of the point measurements is allowed to differ from thevariance of the errors of areal measurements.

In an embodiment, the state equation appears as θ_(t)=θ_(t−1)+v_(t),where v_(t) is a vector of spatially correlated, but temporallyindependent, Gaussian shocks. The state equation describes how thelatent Gaussian processes evolve over time. In order words, the mean andthe variance of this multivariate Gaussian vector depends on time. Theequation above states that, to know the current mean and variance, allthat is needed to be known is the mean and variance of the state at theprevious time step (a so-called first-order Markovian process). In someembodiments, the mean and the Gaussian shocks v_(t) is set to zero,implying that persistence is favored, but the variance is not diagonalmeaning that there is some spatial coherence in the shocks.

In an embodiment, the initial state equation appears as θ₀=m₀+v₀, wherem₀ is a fixed vector of initial means and v₀ is an initial vector ofGaussian shocks. The initial state equation specifies how the model isstarted. For example, if the period under analysis is 200-2016, thisequation describes the mean and variance of the vector of latentGaussian processes in the year 200. In some embodiments, the mean isdefined by considering data that precedes the starting date, so that themodel does not provide unrealistic results as it moves into the firstyears of the period under analysis.

At block 602, the model fitting module 172 sets the initial mean andvariance of the Gaussian processes. In some embodiments, the initialmean and variance are set upon exploratory data analysis or through thefitting of a simple linear regression model to one year of withheld datawhich precedes the period under analysis.

At block 603, the model fitting module 172 updates the mean and varianceof the Gaussian processes for the next time step. In an embodiment, theprocedure used to infer the mean and variance of the latent Gaussianprocesses is a Bayesian generalization of the Kalman Filter. Assume thatthe mean and variance are known for a given time step (which is true forthe initial time step at block 602). The state equation then provides apreliminary update of the mean and variance for the next time step.Using the observation equation, the mean and variance for theobservations at the new time step can be derived both for the point dataand the areal data. The mean and variance for the observations at thenew time step computed via the state equation are then compared to themean and variance of the observations from the observation data 501received at block 600. The mean and variance for the observations at thenew time step computed via the state equation are then updated accordingto Bayes' rule.

At block 604, the model fitting module 172 determines whether the lasttime step has been reached. In an embodiment, the model fitting module172 starts at the first time step of the observation set received atblock 600 and continues until the last time step represented in theobservation set. Therefore, in some embodiments, the model fittingmodule 172 determines whether the last time step represented within theobservation set has been reached, and if so, proceeds to block 605 wherethe fitted model 502 is output to the weather analysis module 173.However, in other embodiments, the fitted model 502 may be stored withinthe model data and field data repository 160 for later retrieval by theweather analysis module 173. If the last time step has not been reached,the model fitting module 172 returns to block 603 to perform the updatefor the next time step.

6.0 Prediction Process Flow

One potential use of the fitted model is to determine a predicted valueof the environmental variable for a location/time for which the valuewas previously unknown. For simplicity, the examples provided in thesection pertain to the case where the time at which to predict theenvironmental variable is within the range of 1≤t≤T and thus the timestep belongs to the period under analysis, where T is the final instantunder analysis. However, the location l does not need to belong to thosesurveyed in the initial observations. For times beyond T the one-stepforecast distribution described earlier can be used to forecasttemperature at time T+1. This formula can be used recursively to getestimates for T+2, T+3, and so forth.

FIG. 7 illustrates an example process flow for determining a predictedvalue of the environmental variable based on a fitted model in blockdiagram form according to an embodiment. FIG. 7 represents an example ofthe aforementioned process and may vary between different embodiments.For example, the steps recited in the blocks of FIG. 7 may be dividedout into multiple sub-steps, combined into a smaller set of steps,performed in a different order, and so forth. In order to illustrateclear examples, the following description assumes that the process flowof FIG. 7 is performed by the data blending subsystem 170.

In FIG. 7, at block 700, the weather analysis module 173 receives afitted model 502. In an embodiment, the fitted model 502 is representedby the observation equation, state equation, and initial state equationin addition to the means and variances of the gridded Gaussian processesdetermined when the model was fitted according to process describedabove in relation to FIG. 6. In some embodiments, the weather analysismodel 173 receives the fitted model 502 from the model fitting module172. However, in other embodiments, the model fitting module 172 maystore the fitted model 502 in the model data and field data repository160. In such embodiments, the weather analysis module 173 retrieves thefitted model 502 from the model data and field data repository 160.

At block 701, the weather analysis module 173 receives the location andtime to predict the value of the environmental variable. In someembodiments, the weather analysis module 173 receives the time step andlocation at which to predict a value for the environmental variablebased on user input received from the field manager computing device104. In some embodiments, more than one pair of time step/locations arereceived at block 701. In such cases, block 702 and block 703 arerepeated for each pair in order to obtain the corresponding predictedvalue and variance. In some embodiments, the user input specifieswhether the location is for a point measurement or an areal average.However, in other embodiments whether the location is for a pointmeasurement or an areal average can be inferred by the weather analysismodule 173 based on the form of the input. For example, a location for apoint measurement may be specified as (x,y) coordinates, whereas anareal average may be specified by grid location or a set of pointsforming the corners of the grid over which the areal average will bepredicted.

At block 702, the weather analysis module 173 constructs an auxiliaryvector which relates to the location with the gridded Gaussian processesfor the specified time. Assuming that the mean (m_(t)) and the variance(V_(t)) of the latent Gaussian processes at time t are available, anauxiliary vector f_(t) can be constructed which relates the location lwith the gridded Gaussian processes. The entries in this vector f_(t)are constructed using the same kernels that were employed to fit themodel to the observation data 501. However, the evaluations obtained areunique because the location is one for which an observation was notrecorded. The kernel that is used depends on whether l is a pointlocation or an areal average. The non-zero kernel evaluations are thenmultiplied with the sine and cosine evaluations pertaining to theinstant t. The result is the auxiliary vector f_(t).

At block 703, the weather analysis module 173 generates a predictionbased on the auxiliary vector, the gridded Gaussian processes, andmeasurement error. In an embodiment, after f_(t) is obtained from block702, the weather analysis module 173 calculates the inner product off_(t) and the vector of Gaussian processes θ_(t) and adds measurementerror: f′_(t) θ_(t)+∈_(t)(l). The result yields a new scalar randomvariable, Y _(e) (l) with scalar mean f′_(t)m_(t) and scalar variancef′_(t)V_(t)f_(t)+v². This random variable provides the predicted meanand variance of an environmental variable at a previously unknownspatio-temporal coordinate.

6.0 Example Applications

The techniques described herein can be used to blend point data andareal averages for many different purposes. In some embodiments, thefitted model 502 can be used to generate analyses and/or reanalyses. Asdescribed above, analyses and reanalyses are typically generated bystarting from an initial state of the environmental system and iteratinga physical process over a number of time steps to obtain a value foreach area of a grid. Since the fitted model 502 allows the initial stateto be generated using both point values and areal averages, the initialstate for the analyses/renalayses generated using the fitted model 502incorporates additional information that has been blended and smoothedto provide a better snapshot of the environmental variable(s) at a giveninitial time over an area of space. Thus, an analyses or reanalysesgenerated using the fitted model 502 to determine an initial stateshould ostensibly be more accurate than using point data or arealaverages separately to set the initial state. Furthermore, the fittedmodel 502 may be used to set the initial state of a forecast model or toevaluate the skill of a forecast model. For example, to evaluate skillthe fitted model 502 could be used to generate predicted values andvariances of the environmental variable(s) which can be compared to theresult of the forecast model to determine the accuracy of the forecastmodel.

Additional use cases may also include (1) generating temporally andspatially coherent simulations (e.g. of temperature), which make use ofthe information contained in multiple data sources (i.e. stations andareal averages). The word “coherent” means that spatial and temporalcorrelation are taken into account, (2) interpolating the gridded dataproduct on to finer grids, making use of available point information,(3) detecting and correcting erroneous point measurements, as they canbe compared with the predictive distribution provided by the model, (4)quantifying the uncertainty and/or calibrate numerical model output,based on station measurements, and (5) making short-term forecasts,using the time-series properties of the model.

7.0 Extensions and Alternatives

In the foregoing specification, embodiments have been described withreference to numerous specific details that may vary from implementationto implementation. The specification and drawings are, accordingly, tobe regarded in an illustrative rather than a restrictive sense. The soleand exclusive indicator of the scope of the disclosure, and what isintended by the applicants to be the scope of the disclosure, is theliteral and equivalent scope of the set of claims that issue from thisapplication, in the specific form in which such claims issue, includingany subsequent correction.

8.0 Additional Disclosure

Aspects of the subject matter described herein are set out in thefollowing numbered clauses:

1. A method comprising: receiving a set of observations of anenvironmental variable, wherein: the set of observations includes one ormore point observations and one or more areal average observations, eachpoint observation of the one or more point observations specifies avalue and a variance of the environmental variable at a particularlocation of an area under observation and a time step of a set of timesteps, and each areal average observation of the one or more arealaverage observations specifies a value and a variance for a district ofa grid superimposed over the area under observation and a time step ofthe set of time steps; defining a state-space model that maps the one ormore point observations and the one or more areal average observationsonto a gridded set of latent Gaussian processes and describes how theset of latent Gaussian processes evolves over time; setting an initialmean and an initial variance for each Gaussian process of the set oflatent Gaussian processes in the state-space model based on observationsof the set of observations that relate to an initial time step of theset of time steps; starting with the initial time step of the set oftime steps and ending at a final time step of the set of time steps,fitting the state-space model to the set of observations by, for eachGaussian process of the set of latent Gaussian processes: generating apredicted mean and a predicted variance for a next time step of theGaussian process based on the state-space model, a mean of the Gaussianprocess at a current time step of the set of time steps, and a varianceof the Gaussian process at the current time step; and updating thepredicted mean and the predicted variance based on one or moreobservations of the set of observations that relate to the next timestep, resulting in a mean and a variance of the Gaussian process for thenext time step.

2. The method of Clause 1, wherein updating the predicted mean and thepredicted variance is performed according to Bayes' rule.

3. The method of any of Clauses 1-2, wherein the state-space modelspecifies that the set of latent Gaussian processes is updated at eachtime step by applying a vector of spatially correlated, but temporallyindependent, Gaussian shocks.

4. The method of any of Clauses 1-3, further comprising: receiving alocation and a time step for which to predict a value and a variance ofthe environmental variable; estimating the value and the variance of theenvironmental variable based on the fitted state-space model.

5. The method of Clause 4, wherein estimating the value and the varianceof the environmental variable is performed by: constructing an auxiliaryvector which relates the location to Gaussian processes of the fittedstate-space model for the time step; estimating the value and thevariance of the environmental variable based on the auxiliary vector,the Gaussian processes, and measurement error.

6. The method of any of Clauses 1-5, wherein the one or more pointobservations represent measurements taken from one or more weatherstations and the one or more areal average observations representmeasurements taken from one or more of: one or more reanalysis productsor one or more satellite images.

7. The method of any of Clauses 1-6, wherein the state-space model mapsthe one or more point observations and the one or more areal averageobservations to the set of latent Gaussian processes using a sparsematrix.

8. The method of Clause 7, wherein each entry of the sparse matrix isconstructed by evaluating a kernel function that only takes non-zerovalues within a predefined spatial range.

9. The method of any of Clauses 7-8, wherein the sparse matrix ismultiplied by a plurality of temporal harmonics to capture seasonalvariability.

10. The method of any of Clauses 1-9, further comprising, in response todetermining that a point observation for a particular time step ismissing from the one or more point observations, using a value and avariance of a corresponding areal average observation of the set ofareal average observations instead when fitting the state-space model.

11. The method of any of Clauses 1-10, wherein the state-space modelmaps the one or more point observations and the one or more arealaverage observations onto the gridded set of latent Gaussian processesusing compactly supported kernels which then held sparse covariancematrices.

12. The method of any of Clauses 1-11, wherein the state space modeltakes into one or more of: measurement error of the set of observationsor estimation error of the set of observations.

13. The method of any of Clauses 1-12, wherein fitting the state-spacemodel applies Bayes' rule to update spatial covariance matrices suchthat the spatial covariance matrices remain sparse during a forwardfitting stage and a backward smoothing stage.

14. The method of any of Clauses 1-13, wherein the state-space modelaccounts for a multitude of seasonal signals, as well as spatialcorrelation, and allows for a magnitude of the seasonal signals and thespatial correlation to vary across space.

15. One or more non-transitory computer-readable media storinginstructions that, when executed by one or more computing devices,causes performance of any one of the methods recited in Clauses 1-14.

16. A system comprising one or more computing devices comprisingcomponents, implemented at least partially by computing hardware,configured to implement the steps of any one of the methods recited inClauses 1-14.

What is claimed is:
 1. A method comprising: receiving, from one or morephysical sensors in an area under observation, a set of observations ofan environmental variable, wherein: the set of observations includes oneor more point observations and one or more areal average observations,each point observation of the one or more point observations specifies avalue and a variance of the environmental variable at a particularlocation of the area under observation and a time step of a set of timesteps, and each areal average observation of the one or more arealaverage observations specifies a value and a variance for a district ofa grid superimposed over the area under observation and a time step ofthe set of time steps; defining a state-space model that maps the one ormore point observations and the one or more areal average observationsonto a set of latent Gaussian processes and describes how the set oflatent Gaussian processes evolves over time; setting an initial mean andan initial variance for each Gaussian process of the set of latentGaussian processes in the state-space model based on observations of theset of observations that relate to an initial time step of the set oftime steps; starting with the initial time step of the set of time stepsand ending at a final time step of the set of time steps, fitting thestate-space model to the set of observations by, for each Gaussianprocess of the set of latent Gaussian processes: generating a predictedmean and a predicted variance for a next time step of the Gaussianprocess based on the state-space model, a mean of the Gaussian processat a current time step of the set of time steps, and a variance of theGaussian process at the current time step; and updating the predictedmean and the predicted variance based on one or more observations of theset of observations that relate to the next time step, resulting in amean and a variance of the Gaussian process for the next time step andupdating the state-space model to improve accuracy of a weather forecastfor the area under observation.
 2. The method of claim 1, whereinupdating the predicted mean and the predicted variance is performedaccording to Bayes' rule.
 3. The method of claim 1, wherein thestate-space model specifies that the set of latent Gaussian processes isupdated at each time step by applying a vector of spatially correlated,but temporally independent, Gaussian shocks.
 4. The method of claim 1,further comprising: receiving a location and a time step for which topredict a value and a variance of the environmental variable; estimatingthe value and the variance of the environmental variable based on thestate-space model.
 5. The method of claim 4, wherein estimating thevalue and the variance of the environmental variable is performed by:constructing an auxiliary vector which relates the location to Gaussianprocesses of the state-space model for the time step; estimating thevalue and the variance of the environmental variable based on theauxiliary vector, the Gaussian processes, and measurement error.
 6. Themethod of claim 1, wherein the one or more point observations representmeasurements taken from one or more weather stations and the one or moreareal average observations represent measurements taken from one or moreof: one or more reanalysis products or one or more satellite images. 7.The method of claim 1, wherein the state-space model maps the one ormore point observations and the one or more areal average observationsto the set of latent Gaussian processes using a sparse matrix.
 8. Themethod of claim 7, wherein each entry of the sparse matrix isconstructed by evaluating a kernel function that only takes non-zerovalues within a predefined spatial range.
 9. The method of claim 7,wherein the sparse matrix is multiplied by a plurality of temporalharmonics to capture seasonal variability.
 10. The method of claim 1,further comprising, in response to determining that a point observationfor a particular time step is missing from the one or more pointobservations, using a value and a variance of a corresponding arealaverage observation of the set of areal average observations insteadwhen fitting the state-space model.
 11. The method of claim 1, whereinthe state-space model maps the one or more point observations and theone or more areal average observations onto the set of latent Gaussianprocesses using compactly supported kernels which then held sparsecovariance matrices.
 12. The method of claim 1, wherein the state-spacemodel takes into one or more of: measurement error of the set ofobservations or estimation error of the set of observations.
 13. Themethod of claim 1, wherein fitting the state-space model applies Bayes'rule to update spatial covariance matrices such that the spatialcovariance matrices remain sparse during a forward fitting stage and abackward smoothing stage.
 14. The method of claim 1, wherein thestate-space model accounts for a multitude of seasonal signals, as wellas spatial correlation, and allows for a magnitude of the seasonalsignals and the spatial correlation to vary across space.
 15. Anon-transitory computer-readable storage medium storing one or moreinstructions which, when executed by one or more processors, cause theone or more processors to perform steps comprising: receiving, from oneor more physical sensors in an area under observation, a set ofobservations of an environmental variable, wherein: the set ofobservations includes one or more point observations and one or moreareal average observations, each point observation of the one or morepoint observations specifies a value and a variance of the environmentalvariable at a particular location of the area under observation and atime step of a set of time steps, and each areal average observation ofthe one or more areal average observations specifies a value and avariance for a district of a grid superimposed over the area underobservation and a time step of the set of time steps; defining astate-space model that maps the one or more point observations and theone or more areal average observations onto a set of latent Gaussianprocesses and describes how the set of latent Gaussian processes evolvesover time; setting an initial mean and an initial variance for eachGaussian process of the set of latent Gaussian processes in thestate-space model based on observations of the set of observations thatrelate to an initial time step of the set of time steps; starting withthe initial time step of the set of time steps and ending at a finaltime step of the set of time steps, fitting the state-space model to theset of observations by, for each Gaussian process of the set of latentGaussian processes: generating a predicted mean and a predicted variancefor a next time step of the Gaussian process based on the state-spacemodel, a mean of the Gaussian process at a current time step of the setof time steps, and a variance of the set of latent Gaussian process atthe current time step; and updating the predicted mean and the predictedvariance based on one or more observations of the set of observationsthat relate to the next time step, resulting in a mean and a variance ofthe Gaussian process for the next time step and updating the state-spacemodel to improve accuracy of a weather forecast for the area underobservation.
 16. The non-transitory computer-readable storage medium ofclaim 15, wherein updating the predicted mean and the predicted varianceis performed according to Bayes' rule.
 17. The non-transitorycomputer-readable storage medium of claim 15, wherein the state-spacemodel specifies that the set of latent Gaussian processes is updated ateach time step by applying a vector of spatially correlated, buttemporally independent, Gaussian shocks.
 18. The non-transitorycomputer-readable storage medium of claim 15, wherein the steps furthercomprise: receiving a location and a time step for which to predict avalue and a variance of the environmental variable; estimating the valueand the variance of the environmental variable based on the fittedstate-space model.
 19. The non-transitory computer-readable storagemedium of claim 18, wherein estimating the value and the variance of theenvironmental variable is performed by: constructing an auxiliary vectorwhich relates the location to Gaussian processes of the state-spacemodel for the time step; estimating the value and the variance of theenvironmental variable based on the auxiliary vector, the Gaussianprocesses, and measurement error.
 20. The non-transitorycomputer-readable storage medium of claim 15, wherein the one or morepoint observations represent measurements taken from one or more weatherstations and the one or more areal average observations representmeasurements taken from one or more of: one or more reanalysis productsor one or more satellite images.
 21. The non-transitorycomputer-readable storage medium of claim 15, wherein the state-spacemodel maps the one or more point observations and the one or more arealaverage observations to the set of latent Gaussian processes using asparse matrix.
 22. The non-transitory computer-readable storage mediumof claim 21, wherein each entry of the sparse matrix is constructed byevaluating a kernel function that only takes non-zero values within apredefined spatial range.
 23. The non-transitory computer-readablestorage medium of claim 21, wherein the sparse matrix is multiplied by aplurality of temporal harmonics to capture seasonal variability.
 24. Thenon-transitory computer-readable storage medium of claim 15, wherein thesteps further comprise: in response to determining that a pointobservation for a particular time step is missing from the one or morepoint observations, using a value and a variance of a correspondingareal average observation of the set of areal average observationsinstead when fitting the state-space model.
 25. The non-transitorycomputer-readable storage medium of claim 15, wherein the state-spacemodel maps the one or more point observations and the one or more arealaverage observations onto the set of latent Gaussian processes usingcompactly supported kernels which then held sparse covariance matrices.26. The non-transitory computer-readable storage medium of claim 15,wherein the state-space model takes into one or more of: measurementerror of the set of observations or estimation error of the set ofobservations.
 27. The non-transitory computer-readable storage medium ofclaim 15, wherein fitting the state-space model applies Bayes' rule toupdate spatial covariance matrices such that the spatial covariancematrices remain sparse during a forward fitting stage and a backwardsmoothing stage.
 28. The non-transitory computer-readable storage mediumof claim 15, wherein the state-space model accounts for a multitude ofseasonal signals, as well as spatial correlation, and allows for amagnitude of the seasonal signals and the spatial correlation to varyacross space.
 29. A system comprising: one or more processors; one ormore non-transitory computer-readable storage mediums storing one ormore instructions which, when executed by the one or more processors,cause the one or more processors to perform steps comprising: receiving,from one or more physical sensors in an area under observation, a set ofobservations of an environmental variable, wherein: the set ofobservations includes one or more point observations and one or moreareal average observations, each point observation of the one or morepoint observations specifies a value and a variance of the environmentalvariable at a particular location of an area under observation and atime step of a set of time steps, and each areal average observation ofthe one or more areal average observations specifies a value and avariance for a district of a grid superimposed over the area underobservation and a time step of the set of time steps; defining astate-space model that maps the one or more point observations and theone or more areal average observations onto a set of latent Gaussianprocesses and describes how the set of latent Gaussian processes evolvesover time; setting an initial mean and an initial variance for eachGaussian process of the set of latent Gaussian processes in thestate-space model based on observations of the set of observations thatrelate to an initial time step of the set of time steps; starting withthe initial time step of the set of time steps and ending at a finaltime step of the set of time steps, fitting the state-space model to theset of observations by, for each Gaussian process of the set of latentGaussian processes: generating a predicted mean and a predicted variancefor a next time step of the Gaussian process based on the state-spacemodel, a mean of the Gaussian process at a current time step of the setof time steps, and a variance of the Gaussian process at the currenttime step; and updating the predicted mean and the predicted variancebased on one or more observations of the set of observations that relateto the next time step, resulting in a mean and a variance of theGaussian process for the next time step and updating the state-spacemodel to improve accuracy of a weather forecast for the area underobservation.